An eigenvector of a square matrix $A$ is a nonzero vector $v$ that, when the matrix is multiplied by $v$, yields a constant multiple of $v$, the multiplier being commonly denoted by $\backslash lambda$. That is:
$A\; v\; =\; \backslash lambda\; v$
(Because this equation uses postmultiplication by $v$, it describes a right eigenvector.)
The number $\backslash lambda$ is called the eigenvalue of $A$ corresponding to $v$.^{[1]}
In analytic geometry, for example, a threeelement vector may be seen as an arrow in threedimensional space starting at the origin. In that case, an eigenvector $v$ is an arrow whose direction is either preserved or exactly reversed after multiplication by $A$. The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.
In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors $\backslash mathbb\{R\}^n$ is replaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), and matrix multiplication is replaced by any linear operator that maps vectors to vectors (such as the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function $f(x)\; =\; a^x$ is an eigenfunction of the derivative operator " $\{\}\text{'}$ ", with eigenvalue $\backslash lambda\; =\; \backslash ln\; a$, since its derivative is $f\text{'}(x)\; =\; (\backslash ln\; a)a^x\; =\; \backslash lambda\; f(x)$.
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix.^{[2]} Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace of a matrix $A$ is the set of all eigenvectors with the same eigenvalue, together with the zero vector.^{[1]} An eigenbasis for $A$ is any basis for the set of all vectors that consists of linearly independent eigenvectors of $A$. Not every matrix has an eigenbasis, but every symmetric matrix does.
The terms characteristic vector, characteristic value, and characteristic space are also used for these concepts. The prefix eigen is adopted from the German word eigen for "self" or "unique to", "peculiar to", or "belonging to"
Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, in quantum mechanics, and in many other areas.
Definition
Eigenvectors and eigenvalues of a real matrix
In many contexts, a vector can be assumed to be a list of real numbers (called elements), written vertically with brackets around the entire list, such as the vectors u and v below. Two vectors are said to be scalar multiples of each other (also called parallel or collinear) if they have the same number of elements, and if every element of one vector is obtained by multiplying each corresponding element in the other vector by the same number (known as a scaling factor, or a scalar). For example, the vectors
 $u\; =\; \backslash begin\{bmatrix\}1\backslash \backslash 3\backslash \backslash 4\backslash end\{bmatrix\}\backslash quad\backslash quad\backslash quad$ and $\backslash quad\backslash quad\backslash quad\; v\; =\; \backslash begin\{bmatrix\}20\backslash \backslash 60\backslash \backslash 80\backslash end\{bmatrix\}$
are scalar multiples of each other, because each element of $v$ is −20 times the corresponding element of $u$.
A vector with three elements, like $u$ or $v$ above, may represent a point in threedimensional space, relative to some Cartesian coordinate system. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition "$u$ is parallel to $v$" means that the two arrows lie on the same straight line, and may differ only in length and direction along that line.
If we multiply any square matrix $A$ with $n$ rows and $n$ columns by such a vector $v$, the result will be another vector $w\; =\; A\; v$, also with $n$ rows and one column. That is,
 $\backslash begin\{bmatrix\}\; v\_1\; \backslash \backslash \; v\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; v\_n\; \backslash end\{bmatrix\}\; \backslash quad\backslash quad$ is mapped to $$
\begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} \;=\;
\begin{bmatrix} A_{1,1} & A_{1,2} & \ldots & A_{1,n} \\
A_{2,1} & A_{2,2} & \ldots & A_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n,1} & A_{n,2} & \ldots & A_{n,n} \\
\end{bmatrix}
\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}
where, for each index $i$,
 $w\_i\; =\; A\_\{i,1\}\; v\_1\; +\; A\_\{i,2\}\; v\_2\; +\; \backslash cdots\; +\; A\_\{i,n\}\; v\_n\; =\; \backslash sum\_\{j\; =\; 1\}^\{n\}\; A\_\{i,j\}\; v\_j$
In general, if $v$ is not all zeros, the vectors $v$ and $A\; v$ will not be parallel. When they are parallel (that is, when there is some real number $\backslash lambda$ such that $A\; v\; =\; \backslash lambda\; v$) we say that $v$ is an eigenvector of $A$. In that case, the scale factor $\backslash lambda$ is said to be the eigenvalue corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix $A$ may change both the direction and the magnitude of an arrow $v$ in threedimensional space. However, if $v$ is an eigenvector of $A$ with eigenvalue $\backslash lambda$, the operation may only change its length, and either keep its direction or flip it (make the arrow point in the exact opposite direction). Specifically, the length of the arrow will increase if $\backslash lambda\; >\; 1$, remain the same if $\backslash lambda\; =\; 1$, and decrease it if $\backslash lambda<\; 1$. Moreover, the direction will be precisely the same if $\backslash lambda\; >\; 0$, and flipped if $\backslash lambda\; <\; 0$. If $\backslash lambda\; =\; 0$, then the length of the arrow becomes zero.
An example
For the transformation matrix
 $A\; =\; \backslash begin\{bmatrix\}\; 3\; \&\; 1\backslash \backslash 1\; \&\; 3\; \backslash end\{bmatrix\},$
the vector
 $v\; =\; \backslash begin\{bmatrix\}\; 4\; \backslash \backslash \; 4\; \backslash end\{bmatrix\}$
is an eigenvector with eigenvalue 2. Indeed,
 $A\; v\; =\; \backslash begin\{bmatrix\}\; 3\; \&\; 1\backslash \backslash 1\; \&\; 3\; \backslash end\{bmatrix\}\; \backslash begin\{bmatrix\}\; 4\; \backslash \backslash \; 4\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; 3\; \backslash cdot\; 4\; +\; 1\; \backslash cdot\; (4)\; \backslash \backslash \; 1\; \backslash cdot\; 4\; +\; 3\; \backslash cdot\; (4)\; \backslash end\{bmatrix\}$
 $=\; \backslash begin\{bmatrix\}\; 8\; \backslash \backslash \; 8\; \backslash end\{bmatrix\}\; =\; 2\; \backslash cdot\; \backslash begin\{bmatrix\}\; 4\; \backslash \backslash \; 4\; \backslash end\{bmatrix\}.$
On the other hand the vector
 $v\; =\; \backslash begin\{bmatrix\}\; 0\; \backslash \backslash \; 1\; \backslash end\{bmatrix\}$
is not an eigenvector, since
 $\backslash begin\{bmatrix\}\; 3\; \&\; 1\backslash \backslash 1\; \&\; 3\; \backslash end\{bmatrix\}\; \backslash begin\{bmatrix\}\; 0\; \backslash \backslash \; 1\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; 3\; \backslash cdot\; 0\; +\; 1\; \backslash cdot\; 1\; \backslash \backslash \; 1\; \backslash cdot\; 0\; +\; 3\; \backslash cdot\; 1\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; 1\; \backslash \backslash \; 3\; \backslash end\{bmatrix\},$
and this vector is not a multiple of the original vector $v$.
Another example
For the matrix
 $A=\; \backslash begin\{bmatrix\}\; 1\; \&\; 1\; \&\; 0\backslash \backslash 0\; \&\; 2\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 3\backslash end\{bmatrix\},$
we have
 $A\; \backslash begin\{bmatrix\}\; 1\backslash \backslash 0\backslash \backslash 0\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; 1\backslash \backslash 0\backslash \backslash 0\; \backslash end\{bmatrix\}\; =\; 1\; \backslash cdot\; \backslash begin\{bmatrix\}\; 1\backslash \backslash 0\backslash \backslash 0\; \backslash end\{bmatrix\},\backslash quad\backslash quad$
 $A\; \backslash begin\{bmatrix\}\; 0\backslash \backslash 0\backslash \backslash 1\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; 0\backslash \backslash 0\backslash \backslash 3\; \backslash end\{bmatrix\}\; =\; 3\; \backslash cdot\; \backslash begin\{bmatrix\}\; 0\backslash \backslash 0\backslash \backslash 1\; \backslash end\{bmatrix\},\backslash quad\backslash quad$
 $A\; \backslash begin\{bmatrix\}\; 1\backslash \backslash 1\backslash \backslash 0\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; 2\backslash \backslash 2\backslash \backslash 0\; \backslash end\{bmatrix\}\; =\; 2\; \backslash cdot\; \backslash begin\{bmatrix\}\; 1\backslash \backslash 1\backslash \backslash 0\; \backslash end\{bmatrix\}.$
Therefore, the vectors $[1,0,0]^\backslash mathsf\{T\}$, $[0,0,1]^\backslash mathsf\{T\}$ and $[1,1,0]^\backslash mathsf\{T\}$ are eigenvectors of $A$ corresponding to the eigenvalues 1, 3, and 2, respectively. (Here the symbol $\{\}^\backslash mathsf\{T\}$ indicates matrix transposition, in this case turning the row vectors into column vectors.)
Trivial cases
The identity matrix $I$ (whose general element $I\_\{i\; j\}$ is 1 if $i\; =\; j$, and 0 otherwise) maps every vector to itself. Therefore, every vector is an eigenvector of $I$, with eigenvalue 1.
More generally, if $A$ is a diagonal matrix (with $A\_\{i\; j\}\; =\; 0$ whenever $i\; \backslash neq\; j$), and $v$ is a vector parallel to axis $i$ (that is, $v\_i\; \backslash neq\; 0$, and $v\_j\; =\; 0$ if $j\; \backslash neq\; i$), then $A\; v\; =\; \backslash lambda\; v$ where $\backslash lambda\; =\; A\_\{i\; i\}$. That is, the eigenvalues of a diagonal matrix are the elements of its main diagonal. This is trivially the case of any 1 ×1 matrix.
General definition
The concept of eigenvectors and eigenvalues extends naturally to abstract linear transformations on abstract vector spaces. Namely, let $V$ be any vector space over some field $K$ of scalars, and let $T$ be a linear transformation mapping $V$ into $V$. We say that a nonzero vector $v$ of $V$ is an eigenvector of $T$ if (and only if) there is a scalar $\backslash lambda$ in $K$ such that
 $T(v)=\backslash lambda\; v$.
This equation is called the eigenvalue equation for $T$, and the scalar $\backslash lambda$ is the eigenvalue of $T$ corresponding to the eigenvector $v$. Note that $T(v)$ means the result of applying the operator $T$ to the vector $v$, while $\backslash lambda\; v$ means the product of the scalar $\backslash lambda$ by $v$.^{[3]}
The matrixspecific definition is a special case of this abstract definition. Namely, the vector space $V$ is the set of all column vectors of a certain size $n$×1, and $T$ is the linear transformation that consists in multiplying a vector by the given $n\backslash times\; n$ matrix $A$.
Some authors allow $v$ to be the zero vector in the definition of eigenvector.^{[4]} This is reasonable as long as we define eigenvalues and eigenvectors carefully: If we would like the zero vector to be an eigenvector, then we must first define an eigenvalue of $T$ as a scalar $\backslash lambda$ in $K$ such that there is a nonzero vector $v$ in $V$ with $T(v)\; =\; \backslash lambda\; v$. We then define an eigenvector to be a vector $v$ in $V$ such that there is an eigenvalue $\backslash lambda$ in $K$ with $T(v)\; =\; \backslash lambda\; v$. This way, we ensure that it is not the case that every scalar is an eigenvalue corresponding to the zero vector.
Eigenspace and spectrum
If $v$ is an eigenvector of $T$, with eigenvalue $\backslash lambda$, then any scalar multiple $\backslash alpha\; v$ of $v$ with nonzero $\backslash alpha$ is also an eigenvector with eigenvalue $\backslash lambda$, since $T(\backslash alpha\; v)\; =\; \backslash alpha\; T(v)\; =\; \backslash alpha(\backslash lambda\; v)\; =\; \backslash lambda(\backslash alpha\; v)$. Moreover, if $u$ and $v$ are eigenvectors with the same eigenvalue $\backslash lambda$, then $u+v$ is also an eigenvector with the same eigenvalue $\backslash lambda$. Therefore, the set of all eigenvectors with the same eigenvalue $\backslash lambda$, together with the zero vector, is a linear subspace of $V$, called the eigenspace of $T$ associated to $\backslash lambda$.^{[5]}^{[6]} If that subspace has dimension 1, it is sometimes called an eigenline.^{[7]}
The geometric multiplicity $\backslash gamma\_T(\backslash lambda)$ of an eigenvalue $\backslash lambda$ is the dimension of the eigenspace associated to $\backslash lambda$, i.e. number of linearly independent eigenvectors with that eigenvalue. These eigenvectors can be chosen so that they are pairwise orthogonal and have unit length under some arbitrary inner product defined on $V$. In other words, every eigenspace has an orthonormal basis of eigenvectors.
Conversely, any eigenvector with eigenvalue $\backslash lambda$ must be linearly independent from all eigenvectors that are associated to a different eigenvalue $\backslash lambda\text{'}$. Therefore a linear transformation $T$ that operates on an $n$dimensional space cannot have more than $n$ distinct eigenvalues (or eigenspaces).^{[8]}
Any subspace spanned by eigenvectors of $T$ is an invariant subspace of $T$.
The list of eigenvalues of $T$ is sometimes called the spectrum of $T$. The order of this list is arbitrary, but the number of times that an eigenvalue $\backslash lambda$ appears is important.
There is no unique way to choose a basis for an eigenspace of an abstract linear operator $T$ based only on $T$ itself, without some additional data such as a choice of coordinate basis for $V$. Even for an eigenline, the basis vector is indeterminate in both magnitude and orientation. If the scalar field $K$ is the real numbers $\backslash mathbb\{R\}$, one can order the eigenspaces by their eigenvalues. Since the modulus $\backslash lambda$ of an eigenvalue is important in many applications, the eigenspaces are often ordered by that criterion.
Eigenbasis
An eigenbasis for a linear operator $T$ that operates on a vector space $V$ is a basis for $V$ that consists entirely of eigenvectors of $T$ (possibly with different eigenvalues). Such a basis may not exist.
Suppose $V$ has finite dimension $n$, and let $\backslash boldsymbol\{\backslash gamma\}\_T$ be the sum of the geometric multiplicities $\backslash gamma\_T(\backslash lambda\_i)$ over all distinct eigenvalues $\backslash lambda\_i$ of $T$. This integer is the maximum number of linearly independent eigenvectors of $T$, and therefore cannot exceed $n$. If $\backslash boldsymbol\{\backslash gamma\}\_T$ is exactly $n$, then $T$ admits an eigenbasis; that is, there exists a basis for $V$ that consists of $n$ eigenvectors. The matrix $A$ that represents $T$ relative to this basis is a diagonal matrix, whose diagonal elements are the eigenvalues associated to each basis vector.
Conversely, if the sum $\backslash boldsymbol\{\backslash gamma\}\_T$ is less than $n$, then $T$ admits no eigenbasis, and there is no choice of coordinates that will allow $T$ to be represented by a diagonal matrix.
Note that $\backslash boldsymbol\{\backslash gamma\}\_T$ is at least equal to the number of distinct eigenvalues of $T$, but may be larger than that.^{[9]} For example, the identity operator $I$ on $V$ has $\backslash boldsymbol\{\backslash gamma\}\_I\; =\; n$, and any basis of $V$ is an eigenbasis of $I$; but its only eigenvalue is 1, with $\backslash gamma\_T(1)\; =\; n$.
Generalizations to infinitedimensional spaces
The definition of eigenvalue of a linear transformation $T$ remains valid even if the underlying space $V$ is an infinite dimensional Hilbert or Banach space. Namely, a scalar $\backslash lambda$ is an eigenvalue if and only if there is some nonzero vector $v$ such that $T(v)\; =\; \backslash lambda\; v$.
Eigenfunctions
A widely used class of linear operators acting on infinite dimensional spaces are the differential operators on function spaces. Let $D$ be a linear differential operator in on the space $\backslash mathbf\{C^\backslash infty\}$ of infinitely differentiable real functions of a real argument $t$. The eigenvalue equation for $D$ is the differential equation
 $D\; f\; =\; \backslash lambda\; f$
The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator $d/dt$, an eigenfunction is a function that, when differentiated, yields a constant times the original function. If $\backslash lambda$ is zero, the generic solution is a constant function. If $\backslash lambda$ is nonzero, the solution is an exponential function
 $f(t)\; =\; Ae^\{\backslash lambda\; t\}.\backslash $
Eigenfunctions are an essential tool in the solution of differential equations and many other applied and theoretical fields. For instance, the exponential functions are eigenfunctions of any shift invariant linear operator. This fact is the basis of powerful Fourier transform methods for solving all sorts of problems.
Spectral theory
If $\backslash lambda$ is an eigenvalue of $T$, then the operator $T\backslash lambda\; I$ is not onetoone, and therefore its inverse $(T\backslash lambda\; I)^\{1\}$ is not defined. The converse is true for finitedimensional vector spaces, but not for infinitedimensional ones. In general, the operator $T\; \; \backslash lambda\; I$ may not have an inverse, even if $\backslash lambda$ is not an eigenvalue.
For this reason, in functional analysis one defines the spectrum of a linear operator $T$ as the set of all scalars $\backslash lambda$ for which the operator $T\backslash lambda\; I$ has no bounded inverse. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
Associative algebras and representation theory
More algebraically, rather than generalizing the vector space to an infinite dimensional space, one can generalize the algebraic object that is acting on the space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
A closer analog of eigenvalues is given by the representationtheoretical concept of weight, with the analogs of eigenvectors and eigenspaces being weight vectors and weight spaces.
Eigenvalues and eigenvectors of matrices
Characteristic polynomial
The eigenvalue equation for a matrix $A$ is
 $A\; v\; \; \backslash lambda\; v\; =\; 0,$
which is equivalent to
 $(A\backslash lambda\; I)v\; =\; 0,$
where $I$ is the $n\backslash times\; n$ identity matrix. It is a fundamental result of linear algebra that an equation $M\; v\; =\; 0$ has a nonzero solution $v$ if, and only if, the determinant $\backslash det(M)$ of the matrix $M$ is zero. It follows that the eigenvalues of $A$ are precisely the real numbers $\backslash lambda$ that satisfy the equation
 $\backslash det(A\backslash lambda\; I)\; =\; 0$
The lefthand side of this equation can be seen (using Leibniz' rule for the determinant) to be a polynomial function of the variable $\backslash lambda$. The degree of this polynomial is $n$, the order of the matrix. Its coefficients depend on the entries of $A$, except that its term of degree $n$ is always $(1)^n\backslash lambda^n$. This polynomial is called the characteristic polynomial of $A$; and the above equation is called the characteristic equation (or, less often, the secular equation) of $A$.
For example, let $A$ be the matrix
 $A\; =$
\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 & 4 \\
0 & 4 & 9
\end{bmatrix}
The characteristic polynomial of $A$ is
 $\backslash det\; (A\backslash lambda\; I)\; \backslash ;=\backslash ;\; \backslash det\; \backslash left(\backslash begin\{bmatrix\}$
2 & 0 & 0 \\
0 & 3 & 4 \\
0 & 4 & 9
\end{bmatrix}  \lambda
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}\right) \;=\;
\det \begin{bmatrix}
2  \lambda & 0 & 0 \\
0 & 3  \lambda & 4 \\
0 & 4 & 9  \lambda
\end{bmatrix}
which is
 $(2\; \; \backslash lambda)\; \backslash bigl[\; (3\; \; \backslash lambda)\; (9\; \; \backslash lambda)\; \; 16\; \backslash bigr]\; =\; \backslash lambda^3\; +\; 14\backslash lambda^2\; \; 35\backslash lambda\; +\; 22$
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of $A$, corresponding to the eigenvectors $[1,0,0]\text{'},$ $[0,2,1]\text{'},$ and $[0,1,2]\text{'}$ (or any nonzero multiples thereof).
In the real domain
Since the eigenvalues are roots of the characteristic polynomial, an $n\backslash times\; n$ matrix has at most $n$ eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real; but it may have fewer than $n$ real roots, or no real roots at all.
For example, consider the cyclic permutation matrix
 $A\; =\; \backslash begin\{bmatrix\}\; 0\; \&\; 1\; \&\; 0\backslash \backslash 0\; \&\; 0\; \&\; 1\backslash \backslash \; 1\; \&\; 0\; \&\; 0\backslash end\{bmatrix\}$
This matrix shifts the coordinates of the vector up by one position, and moves the first coordinate to the bottom. Its characteristic polynomial is $1\; \; \backslash lambda^3$ which has one real root $\backslash lambda\_1\; =\; 1$. Any vector with three equal nonzero elements is an eigenvector for this eigenvalue. For example,
 $$
A \begin{bmatrix} 5\\5\\5 \end{bmatrix} =
\begin{bmatrix} 5\\5\\5 \end{bmatrix} =
1 \cdot \begin{bmatrix} 5\\5\\5 \end{bmatrix}
In the complex domain
The fundamental theorem of algebra implies that the characteristic polynomial of an $n\backslash times\; n$ matrix $A$, being a polynomial of degree $n$, has exactly $n$ complex roots. More precisely, it can be factored into the product of $n$ linear terms,
 $\backslash det(A\backslash lambda\; I)\; =\; (\backslash lambda\_1\; \; \backslash lambda\; )(\backslash lambda\_2\; \; \backslash lambda)\backslash cdots(\backslash lambda\_n\; \; \backslash lambda)$
where each $\backslash lambda\_i$ is a complex number. The numbers $\backslash lambda\_1$, $\backslash lambda\_2$, ... $\backslash lambda\_n$, (which may not be all distinct) are roots of the polynomial, and are precisely the eigenvalues of $A$.
Even if the entries of $A$ are all real numbers, the eigenvalues may still have nonzero imaginary parts (and the elements of the corresponding eigenvectors will therefore also have nonzero imaginary parts). Also, the eigenvalues may be irrational numbers even if all the entries of $A$ are rational numbers, or all are integers. However, if the entries of $A$ are algebraic numbers (which include the rationals), the eigenvalues will be (complex) algebraic numbers too.
The nonreal roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugate values, namely with the two members of each pair having the same real part and imaginary parts that differ only in sign. If the degree is odd, then by the intermediate value theorem at least one of the roots will be real. Therefore, any real matrix with odd order will have at least one real eigenvalue; whereas a real matrix with even order may have no real eigenvalues.
In the example of the 3×3 cyclic permutation matrix $A$, above, the characteristic polynomial $1\; \; \backslash lambda^3$ has two additional nonreal roots, namely
 $\backslash lambda\_2\; =\; 1/2\; +\; \backslash mathbf\{i\}\backslash sqrt\{3\}/2\backslash quad\backslash quad$ and $\backslash quad\backslash quad\backslash lambda\_3\; =\; \backslash lambda\_2^*\; =\; 1/2\; \; \backslash mathbf\{i\}\backslash sqrt\{3\}/2$,
where $\backslash mathbf\{i\}=\; \backslash sqrt\{1\}$ is the imaginary unit. Note that $\backslash lambda\_2\backslash lambda\_3\; =\; 1$, $\backslash lambda\_2^2\; =\; \backslash lambda\_3$, and $\backslash lambda\_3^2\; =\; \backslash lambda\_2$. Then
 $$
A \begin{bmatrix} 1 \\ \lambda_2 \\ \lambda_3 \end{bmatrix} =
\begin{bmatrix} \lambda_2\\ \lambda_3 \\1 \end{bmatrix} =
\lambda_2 \cdot \begin{bmatrix} 1\\ \lambda_2 \\ \lambda_3 \end{bmatrix}
\quad\quad
and $\backslash quad\backslash quad\; A\; \backslash begin\{bmatrix\}\; 1\; \backslash \backslash \; \backslash lambda\_3\; \backslash \backslash \; \backslash lambda\_2\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; \backslash lambda\_3\; \backslash \backslash \; \backslash lambda\_2\; \backslash \backslash \; 1\; \backslash end\{bmatrix\}\; =\; \backslash lambda\_3\; \backslash cdot\; \backslash begin\{bmatrix\}\; 1\; \backslash \backslash \; \backslash lambda\_3\; \backslash \backslash \; \backslash lambda\_2\; \backslash end\{bmatrix\}$
Therefore, the vectors $[1,\backslash lambda\_2,\backslash lambda\_3]\text{'}$ and $[1,\backslash lambda\_3,\backslash lambda\_2]\text{'}$ are eigenvectors of $A$, with eigenvalues $\backslash lambda\_2$, and $\backslash lambda\_3$, respectively.
Algebraic multiplicities
Let $\backslash lambda\_i$ be an eigenvalue of an $n\backslash times\; n$ matrix $A$. The algebraic multiplicity $\backslash mu\_A(\backslash lambda\_i)$ of $\backslash lambda\_i$ is its multiplicity as a root of the characteristic polynomial, that is, the largest integer $k$ such that $(\backslash lambda\; \; \backslash lambda\_i)^k$ divides evenly that polynomial.
Like the geometric multiplicity $\backslash gamma\_A(\backslash lambda\_i)$, the algebraic multiplicity is an integer between 1 and $n$; and the sum $\backslash boldsymbol\{\backslash mu\}\_A$ of $\backslash mu\_A(\backslash lambda\_i)$ over all distinct eigenvalues also cannot exceed $n$. If complex eigenvalues are considered, $\backslash boldsymbol\{\backslash mu\}\_A$ is exactly $n$.
It can be proved that the geometric multiplicity $\backslash gamma\_A(\backslash lambda\_i)$ of an eigenvalue never exceeds its algebraic multiplicity $\backslash mu\_A(\backslash lambda\_i)$. Therefore, $\backslash boldsymbol\{\backslash gamma\}\_A$ is at most $\backslash boldsymbol\{\backslash mu\}\_A$.
If $\backslash gamma\_A(\backslash lambda\_i)\; =\; \backslash mu\_A(\backslash lambda\_i)$, then $\backslash lambda\_i$ is said to be a semisimple eigenvalue.
Example
For the matrix:
$A=\; \backslash begin\{bmatrix\}\; 2\; \&\; 0\; \&\; 0\; \&\; 0\; \backslash \backslash \; 1\; \&\; 2\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 3\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; 3\; \backslash end\{bmatrix\},$
 the characteristic polynomial of $A$ is $\backslash det\; (A\backslash lambda\; I)\; \backslash ;=\backslash ;$
\det \begin{bmatrix}
2 \lambda & 0 & 0 & 0 \\
1 & 2 \lambda & 0 & 0 \\
0 & 1 & 3 \lambda & 0 \\
0 & 0 & 1 & 3 \lambda
\end{bmatrix}= (2  \lambda)^2 (3  \lambda)^2 ,
 being the product of the diagonal with a lower triangular matrix.
The roots of this polynomial, and hence the eigenvalues, are 2 and 3.
The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots.
On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by the vector $[0,1,1,1]$, and is therefore 1 dimensional.
Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by $[0,0,0,1]$. Hence, the total algebraic multiplicity of A, denoted $\backslash mu\_A$, is 4, which is the most it could be for a 4 by 4 matrix. The geometric multiplicity $\backslash gamma\_A$ is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.
Diagonalization and eigendecomposition
If the sum $\backslash boldsymbol\{\backslash gamma\}\_A$ of the geometric multiplicities of all eigenvalues is exactly $n$, then $A$ has a set of $n$ linearly independent eigenvectors. Let $Q$ be a square matrix whose columns are those eigenvectors, in any order. Then we will have $A\; Q\; =\; Q\backslash Lambda$, where $\backslash Lambda$ is the diagonal matrix such that $\backslash Lambda\_\{i\; i\}$ is the eigenvalue associated to column $i$ of $Q$. Since the columns of $Q$ are linearly independent, the matrix $Q$ is invertible. Premultiplying both sides by $Q^\{1\}$ we get $Q^\{1\}A\; Q\; =\; \backslash Lambda$. By definition, therefore, the matrix $A$ is diagonalizable.
Conversely, if $A$ is diagonalizable, let $Q$ be a nonsingular square matrix such that $Q^\{1\}\; A\; Q$ is some diagonal matrix $D$. Multiplying both sides on the left by $Q$ we get $A\; Q\; =\; Q\; D$. Therefore each column of $Q$ must be an eigenvector of $A$, whose eigenvalue is the corresponding element on the diagonal of $D$. Since the columns of $Q$ must be linearly independent, it follows that $\backslash boldsymbol\{\backslash gamma\}\_A\; =\; n$. Thus $\backslash boldsymbol\{\backslash gamma\}\_A$ is equal to $n$ if and only if $A$ is diagonalizable.
If $A$ is diagonalizable, the space of all $n$element vectors can be decomposed into the direct sum of the eigenspaces of $A$. This decomposition is called the eigendecomposition of $A$, and it is the preserved under change of coordinates.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvector can be generalized to generalized eigenvectors, and that of diagonal matrix to a Jordan form matrix. Over an algebraically closed field, any matrix $A$ has a Jordan form and therefore admits a basis of generalized eigenvectors, and a decomposition into generalized eigenspaces
Further properties
Let $A$ be an arbitrary $n\backslash times\; n$ matrix of complex numbers with eigenvalues $\backslash lambda\_1$, $\backslash lambda\_2$, ... $\backslash lambda\_n$. (Here it is understood that an eigenvalue with algebraic multiplicity $\backslash mu$ occurs $\backslash mu$ times in this list.) Then
 The trace of $A$, defined as the sum of its diagonal elements, is also the sum of all eigenvalues:
 $\backslash operatorname\{tr\}(A)\; =\; \backslash sum\_\{i=1\}^n\; A\_\{i\; i\}\; =\; \backslash sum\_\{i=1\}^n\; \backslash lambda\_i\; =\; \backslash lambda\_1+\; \backslash lambda\_2\; +\backslash cdots+\; \backslash lambda\_n$.
 The determinant of $A$ is the product of all eigenvalues:
 $\backslash operatorname\{det\}(A)\; =\; \backslash prod\_\{i=1\}^n\; \backslash lambda\_i=\backslash lambda\_1\backslash lambda\_2\backslash cdots\backslash lambda\_n$.
 The eigenvalues of the $k$th power of $A$, i.e. the eigenvalues of $A^k$, for any positive integer $k$, are $\backslash lambda\_1^k,\backslash lambda\_2^k,\backslash dots,\backslash lambda\_n^k$
 The matrix $A$ is invertible if and only if all the eigenvalues $\backslash lambda\_i$ are nonzero.
 If $A$ is invertible, then the eigenvalues of $A^\{1\}$ are $1/\backslash lambda\_1,1/\backslash lambda\_2,\backslash dots,1/\backslash lambda\_n$
 If $A$ is equal to its conjugate transpose $A^*$ (in other words, if $A$ is Hermitian), then every eigenvalue is real. The same is true of any a symmetric real matrix. If $A$ is also positivedefinite, positivesemidefinite, negativedefinite, or negativesemidefinite every eigenvalue is positive, nonnegative, negative, or nonpositive respectively.
 Every eigenvalue of a unitary matrix has absolute value $\backslash lambda=1$.
Left and right eigenvectors
The use of matrices with a single column (rather than a single row) to represent vectors is traditional in many disciplines. For that reason, the word "eigenvector" almost always means a right eigenvector, namely a column vector that must placed to the right of the matrix $A$ in the defining equation
 $A\; v\; =\; \backslash lambda\; v$.
There may be also singlerow vectors that are unchanged when they occur on the left side of a product with a square matrix $A$; that is, which satisfy the equation
 $u\; A\; =\; \backslash lambda\; u$
Any such row vector $u$ is called a left eigenvector of $A$.
The left eigenvectors of $A$ are transposes of the right eigenvectors of the transposed matrix $A^\backslash mathsf\{T\}$, since their defining equation is equivalent to
 $A^\backslash mathsf\{T\}\; u^\backslash mathsf\{T\}\; =\; \backslash lambda\; u^\backslash mathsf\{T\}$
It follows that, if $A$ is Hermitian, its left and right eigenvectors are complex conjugates. In particular if $A$ is a real symmetric matrix, they are the same except for transposition.
Calculation
Computing the eigenvalues
The eigenvalues of a matrix $A$ can be determined by finding the roots of the characteristic polynomial. Explicit algebraic formulas for the roots of a polynomial exist only if the degree $n$ is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.
It turns out that any polynomial with degree $n$ is the characteristic polynomial of some companion matrix of order $n$. Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.^{[10]} However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable roundoff errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial).^{[10]}
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961.
^{[10]} Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.^{[10]}
Computing the eigenvectors
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
 $A\; =\; \backslash begin\{bmatrix\}\; 4\; \&\; 1\backslash \backslash 6\; \&\; 3\; \backslash end\{bmatrix\}$
we can find its eigenvectors by solving the equation $A\; v\; =\; 6\; v$, that is
 $\backslash begin\{bmatrix\}\; 4\; \&\; 1\backslash \backslash 6\; \&\; 3\; \backslash end\{bmatrix\}\backslash begin\{bmatrix\}x\backslash \backslash y\backslash end\{bmatrix\}\; =\; 6\; \backslash cdot\; \backslash begin\{bmatrix\}x\backslash \backslash y\backslash end\{bmatrix\}$
This matrix equation is equivalent to two linear equations
 $$
\left\{\begin{matrix} 4x + {\ }y &{}= 6x\\6x + 3y &{}=6 y\end{matrix}\right.
\quad\quad\quad that is $\backslash left\backslash \{\backslash begin\{matrix\}\; 2x+\; \{\backslash \; \}y\; \&\{\}=0\backslash \backslash +6x3y\; \&\{\}=0\backslash end\{matrix\}\backslash right.$
Both equations reduce to the single linear equation $y=2x$. Therefore, any vector of the form $[a,2a]\text{'}$, for any nonzero real number $a$, is an eigenvector of $A$ with eigenvalue $\backslash lambda\; =\; 6$.
The matrix $A$ above has another eigenvalue $\backslash lambda=1$. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of $3x+y=0$, that is, any vector of the form $[b,3b]\text{'}$, for any nonzero real number $b$.
Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a byproduct of the computation.
History
Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.^{[11]} In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.^{[12]} Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.^{[13]}
Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur.^{[14]} Sturm developed Fourier's ideas further and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues.^{[12]} This was extended by Hermite in 1855 to what are now called Hermitian matrices.^{[13]} Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,^{[12]} and Clebsch found the corresponding result for skewsymmetric matrices.^{[13]} Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.^{[12]}
In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.^{[15]} Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.^{[16]}
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.^{[17]} He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.^{[18]}
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis^{[19]} and Vera Kublanovskaya^{[20]} in 1961.^{[21]}
Applications
Eigenvalues of geometric transformations
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.

scaling

unequal scaling

rotation

horizontal shear

hyperbolic rotation

illustration






matrix

$\backslash begin\{bmatrix\}k\; \&\; 0\backslash \backslash 0\; \&\; k\backslash end\{bmatrix\}$

$\backslash begin\{bmatrix\}k\_1\; \&\; 0\backslash \backslash 0\; \&\; k\_2\backslash end\{bmatrix\}$

$\backslash begin\{bmatrix\}c\; \&\; s\; \backslash \backslash \; s\; \&\; c\backslash end\{bmatrix\}$ $c=\backslash cos\backslash theta$ $s=\backslash sin\backslash theta$

$\backslash begin\{bmatrix\}1\; \&\; k\backslash \backslash \; 0\; \&\; 1\backslash end\{bmatrix\}$

$\backslash begin\{bmatrix\}\; c\; \&\; s\; \backslash \backslash \; s\; \&\; c\; \backslash end\{bmatrix\}$ $c=\backslash cosh\; \backslash varphi$ $s=\backslash sinh\; \backslash varphi$

characteristic polynomial

$\backslash \; (\backslash lambda\; \; k)^2$

$(\backslash lambda\; \; k\_1)(\backslash lambda\; \; k\_2)$

$\backslash lambda^2\; \; 2c\backslash lambda\; +\; 1$

$\backslash \; (\backslash lambda\; \; 1)^2$

$\backslash lambda^2\; \; 2c\backslash lambda\; +\; 1$

eigenvalues $\backslash lambda\_i$

$\backslash lambda\_1\; =\; \backslash lambda\_2\; =\; k$

$\backslash lambda\_1\; =\; k\_1$ $\backslash lambda\_2\; =\; k\_2$

$\backslash lambda\_1\; =\; e^\{\backslash mathbf\{i\}\backslash theta\}=c+s\backslash mathbf\{i\}$ $\backslash lambda\_2\; =\; e^\{\backslash mathbf\{i\}\backslash theta\}=cs\backslash mathbf\{i\}$

$\backslash lambda\_1\; =\; \backslash lambda\_2\; =\; 1$

$\backslash lambda\_1\; =\; e^\backslash varphi$ $\backslash lambda\_2\; =\; e^\{\backslash varphi\}$,

algebraic multipl. $\backslash mu\_i=\backslash mu(\backslash lambda\_i)$

$\backslash mu\_1\; =\; 2$

$\backslash mu\_1\; =\; 1$ $\backslash mu\_2\; =\; 1$

$\backslash mu\_1\; =\; 1$ $\backslash mu\_2\; =\; 1$

$\backslash mu\_1\; =\; 2$

$\backslash mu\_1\; =\; 1$ $\backslash mu\_2\; =\; 1$

geometric multipl. $\backslash gamma\_i\; =\; \backslash gamma(\backslash lambda\_i)$

$\backslash gamma\_1\; =\; 2$

$\backslash gamma\_1\; =\; 1$ $\backslash gamma\_2\; =\; 1$

$\backslash gamma\_1\; =\; 1$ $\backslash gamma\_2\; =\; 1$

$\backslash gamma\_1\; =\; 1$

$\backslash gamma\_1\; =\; 1$ $\backslash gamma\_2\; =\; 1$

eigenvectors

All nonzero vectors

$u\_1\; =\; \backslash begin\{bmatrix\}1\backslash \backslash 0\backslash end\{bmatrix\}$ $u\_2\; =\; \backslash begin\{bmatrix\}0\backslash \backslash 1\backslash end\{bmatrix\}$

$u\_1\; =\; \backslash begin\{bmatrix\}\{\backslash \; \}1\backslash \backslash \backslash mathbf\{i\}\backslash end\{bmatrix\}$ $u\_2\; =\; \backslash begin\{bmatrix\}\{\backslash \; \}1\backslash \backslash \; +\backslash mathbf\{i\}\backslash end\{bmatrix\}$

$u\_1\; =\; \backslash begin\{bmatrix\}1\backslash \backslash 0\backslash end\{bmatrix\}$

$u\_1\; =\; \backslash begin\{bmatrix\}\{\backslash \; \}1\backslash \backslash \{\backslash \; \}1\backslash end\{bmatrix\}$ $u\_2\; =\; \backslash begin\{bmatrix\}\{\backslash \; \}1\backslash \backslash 1\backslash end\{bmatrix\}.$

Note that the characteristic equation for a rotation is a quadratic equation with discriminant $D\; =\; 4(\backslash sin\backslash theta)^2$, which is a negative number whenever $\backslash theta$ is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, $\backslash cos\backslash theta\; \backslash pm\; \backslash mathbf\{i\}\backslash sin\backslash theta$; and all eigenvectors have nonreal entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
Schrödinger equation
An example of an eigenvalue equation where the transformation $T$ is represented in terms of a differential operator is the timeindependent Schrödinger equation in quantum mechanics:
 $H\backslash psi\_E\; =\; E\backslash psi\_E\; \backslash ,$
where $H$, the Hamiltonian, is a secondorder differential operator and $\backslash psi\_E$, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue $E$, interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for $\backslash psi\_E$ within the space of square integrable functions. Since this space is a Hilbert space with a welldefined scalar product, one can introduce a basis set in which $\backslash psi\_E$ and $H$ can be represented as a onedimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
Braket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by $\backslash Psi\_E\backslash rangle$. In this notation, the Schrödinger equation is:
 $H\backslash Psi\_E\backslash rangle\; =\; E\backslash Psi\_E\backslash rangle$
where $\backslash Psi\_E\backslash rangle$ is an eigenstate of $H$. It is a self adjoint operator, the infinite dimensional analog of Hermitian matrices (see Observable). As in the matrix case, in the equation above $H\backslash Psi\_E\backslash rangle$ is understood to be the vector obtained by application of the transformation $H$ to $\backslash Psi\_E\backslash rangle$.
Molecular orbitals
In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. If one wants to underline this aspect one speaks of nonlinear eigenvalue problem. Such equations are usually solved by an iteration procedure, called in this case selfconsistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a nonorthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations.
Geology and glaciology
In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a TriPlot (Sneed and Folk) diagram,^{[22]}^{[23]} or as a Stereonet on a Wulff Net.^{[24]}
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered $v\_1,\; v\_2,\; v\_3$ by their eigenvalues $E\_1\; \backslash geq\; E\_2\; \backslash geq\; E\_3$;^{[25]} $v\_1$ then is the primary orientation/dip of clast, $v\_2$ is the secondary and $v\_3$ is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of $E\_1$, $E\_2$, and $E\_3$ are dictated by the nature of the sediment's fabric. If $E\_1\; =\; E\_2\; =\; E\_3$, the fabric is said to be isotropic. If $E\_1\; =\; E\_2\; >\; E\_3$, the fabric is said to be planar. If $E\_1\; >\; E\_2\; >\; E\_3$, the fabric is said to be linear.^{[26]}
Principal components analysis
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal components analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthonormal eigenbasis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principalcomponents that are associated with most of the covariability among a number of observed data.
Principal component analysis is used to study large data sets, such as those encountered in data mining, chemical research, psychology, and in marketing. PCA is popular especially in psychology, in the field of psychometrics. In Q methodology, the eigenvalues of the correlation matrix determine the Qmethodologist's judgment of practical significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.
Vibration analysis
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by
 $m\backslash ddot\; x\; +\; kx\; =\; 0$
or
 $m\backslash ddot\; x\; =\; k\; x$
that is, acceleration is proportional to position (i.e., we expect $x$ to be sinusoidal in time).
In $n$ dimensions, $m$ becomes a mass matrix and $k$ a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
 $k\; x\; =\; \backslash omega^2\; m\; x$
where $\backslash omega^2$ is the eigenvalue and $\backslash omega$ is the angular frequency. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of $k$ alone. Furthermore, damped vibration, governed by
 $m\backslash ddot\; x\; +\; c\; \backslash dot\; x\; +\; kx\; =\; 0$
leads to what is called a socalled quadratic eigenvalue problem,
 $(\backslash omega^2\; m\; +\; \backslash omega\; c\; +\; k)x\; =\; 0.$
This can be reduced to a generalized eigenvalue problem by clever use of algebra at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalarvalued vibration problems.
Eigenfaces
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.^{[27]} The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal components analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.
Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems, for speaker adaptation.
Tensor of moment of inertia
In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
Stress tensor
In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.
Eigenvalues of a graph
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix $A$, or (increasingly) of the graph's Laplacian matrix (see also Discrete Laplace operator), which is either $T\; \; A$ (sometimes called the combinatorial Laplacian) or $I\; \; T^\{1/2\}A\; T^\{1/2\}$ (sometimes called the normalized Laplacian), where $T$ is a diagonal matrix with $T\_\{i\; i\}$ equal to the degree of vertex $v\_i$, and in $T^\{1/2\}$, the $i$th diagonal entry is $\backslash sqrt\{\backslash operatorname\{deg\}(v\_i)\}$. The $k$th principal eigenvector of a graph is defined as either the eigenvector corresponding to the $k$th largest or $k$th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the rownormalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
Basic reproduction number
 See Basic reproduction number
The basic reproduction number ($R\_0$) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then $R\_0$ is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, $t\_G$, from one person becoming infected to the next person becoming infected. In a heterogenous population, the next generation matrix defines how many people in the population will become infected after time $t\_G$ has passed. $R\_0$ is then the largest eigenvalue of the next generation matrix.^{[28]}^{[29]}
See also
Notes
References
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 Pigolkina, T. S. and Shulman, V. S., Eigenvalue (in Russian), In:Vinogradov, I. M. (Ed.), Mathematical Encyclopedia, Vol. 5, Soviet Encyclopedia, Moscow, 1977.
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 Curtis, Charles W., Linear Algebra: An Introductory Approach, 347 p., Springer; 4th ed. 1984. Corr. 7th printing edition (August 19, 1999), ISBN 0387909923.
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External links
 What are Eigen Values? – nontechnical introduction from PhysLink.com's "Ask the Experts"
 Eigen Values and Eigen Vectors Numerical Examples – Tutorial and Interactive Program from Revoledu.
 Introduction to Eigen Vectors and Eigen Values – lecture from Khan Academy

Theory
 Template:Springer
 Template:Springer
 Template:Planetmath reference
 MathWorld
 Eigen Vector Examination working applet
 Same Eigen Vector Examination as above in a Flash demo with sound
 Computation of Eigenvalues
 Henk van der Vorst
 Eigenvalues and Eigenvectors on the Ask Dr. Math forums: [2]
Online calculators
 arndtbruenner.de
 bluebit.gr
 wims.unice.fr
Demonstration applets
 Java applet about eigenvectors in the real plane
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