Let be a square, symmetric matrix. A real scalar is said to be an eigenvalue of if there exist a non-zero vector such that
The vector is then referred to as an eigenvector associated with the eigenvalue . The eigenvector is said to be normalized if . In this case, we have
The interpretation of is that it defines a direction along behaves just like scalar multiplication. The amount of scaling is given by . (In German, the root ‘‘eigen’’, means ‘‘self’’ or ‘‘proper’’). The eigenvalues of the matrix are characterized by the characteristic equation
where the notation refers to the determinant of its matrix argument. The function with values is a polynomial of degree called the characteristic polynomial.
From the fundamental theorem of algebra, any polynomial of degree has (possibly not distinct) complex roots. For symmetric matrices, the eigenvalues are real, since when , and is normalized.
An important result of linear algebra, called the spectral theorem, or symmetric eigenvalue decomposition (SED) theorem, states that for any symmetric matrix, there are exactly (possibly not distinct) eigenvalues, and they are all real; further, that the associated eigenvectors can be chosen so as to form an orthonormal basis. The result offers a simple way to decompose the symmetric matrix as a product of simple transformations.
Theorem: Symmetric eigenvalue decomposition (Proof)
We can decompose any symmetric matrix with the symmetric eigenvalue decomposition (SED)
where the matrix of is orthogonal (that is, ), and contains the eigenvectors of , while the diagonal matrix contains the eigenvalues of .
The SED provides a decomposition of the matrix in simple terms, namely dyads. We check that in the SED above, the scalars are the eigenvalues, and 's are associated eigenvectors, since
The eigenvalue decomposition of a symmetric matrix can be efficiently computed with standard software, in time that grows proportionately to its dimension as . Here is the matlab syntax, where the first line ensure that matlab knows that the matrix is exactly symmetric.Matlab syntax
>> A = triu(A)+tril(A',-1); >> [U,D] = eig(A);
Given a symmetric matrix , we can express the smallest and largest eigenvalues of , denoted and respectively, in the so-called variational form
For a proof, see here.
The term ‘‘variational’’ refers to the fact that the eigenvalues are given as optimal values of optimization problems, which were referred to in the past as variational problems. Variational representations exist for all the eigenvalues, but are more complicated to state.
The interpretation of the above identities is that the largest and smallest eigenvalues is a measure of the range of the quadratic function over the unit Euclidean ball. The quantities above can be written as the minimum and maximum of the so-called Rayleigh quotient.
Historically, David Hilbert coined the term ‘‘spectrum’’ for the set of eigenvalues of a symmetric operator (roughly, a matrix of infinite dimensions). The fact that for symmetric matrices, every eigenvalue lies in the interval somewhat justifies the terminology.
Example: Largest singular value norm of a matrix.