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The eigenvalues of a matrix are the roots of its characteristic equation.
They may also be referred to by any of the fourteen other combinations of:
[characteristic, eigen, latent, proper, secular] + [number, root, value].
An eigenvalue c has algebraic multiplicity k if
(t-c)k is the highest power of (t-c) that
divides the characteristic polynomial.
[n*n] The characteristic equation of a matrix
A is |tI-A| = 0. It is a polynomial equation in
- [n*n] A matrix A satisfies its own
characteristic equation (Cayley-Hamilton theorem)
[n*n]: The characteristic matrix of A is
(tI-A) and is a function of the scalar t.
[n*n] The characteristic polynomial,
p(t), of a matrix A is p(t) = |tI -
- [n*n]: The characteristic polynomial of
A is of the form: tn -
tr(A)*tn-1 + ... + -1n
- [2*2]: |tI-A| =
t2 - tr(A)*t + |A|
- [A,B: m*n]: If m>n
|tI - AB'| = tm-n * |tI -
- [n*n]: |tI-AB| =
An eigenvalue is defective if its geometric
multiplicity is less than its algebraic
The eigenvalues of A are the roots of its characteristic equation: |tI-A| = 0.
The function eig(A) denotes a column vector containing all the
eigenvalues of A with appropriate multiplicities.
- t is an eigenvalue of A:n*n iff for some non-zero x,
Ax=tx. x is then called an eigenvector corresponding to
- [Complex, n*n]: The matrix A has
exactly n eigenvalues (not necessarily distinct)
- [Complex]: tr(A) =
- [Complex]: det(A) =
- [A:m*m, C:n*n]: eig([A B; 0 C]) =
- det(A)=0 iff 0 is an eigenvalue of A
- The eigenvalues of a triangular or diagonal matrix are its diagonal
- [Hermitian]: The eigenvalues of A are
The eigenvalues of A have unit modulus.
- [Nilpotent]: The eigenvalues of A are
- [Idempotent]: The eigenvalues of A
are all either 0 or 1.
- The eigenvalues of Ak are
- Similar matrices have the same
- [Real, Symmetric, 2#2] A=[a
b; b d]=RDRT where R=[cos(t)
-sin(t); sin(t) cos(t)] and
t=½tan-1(2b/(a-d)) and D is
diagonal with D=DIAG([ 1 cos(2t) sin(2t); 1
-cos(2t) -sin(2t)] *[ (a+d)/2;
(a-d)/2; b]). The columns of R are the eigenvectors
and the corresponding elements of D the eigenvalues.
The geometric multiplicity of an eigenvalue c of a matrix A is the dimension
of the subspace of vectors x for which Ax = cx.
The eigenvalues of a diagonal matrix equal its diagonal elements. If the
off-diagonal elements are small rather than being exactly zero, the eigenvalues
will be close to the diagonal elements; Gersgorin Discs make this statement
- The eigenvalues of A[n#n] lie in the union
of the n complex-plane closed discs whose centres are
diag(A) and whose radii are
sum(ABS(A))-diag(ABS(A)). These discs
are the Gersgorin discs (there should be a circumflex over the s of Gersgorin)
- Each eigenvalue, c, satisfies min(2
diag(ABS(A)) - sum(ABS(A)))
<= |c| <= max(sum(ABS(A)))
- If the n discs can be partitioned into disjoint subsets of the
complex plane then each subset contains the same number of (not necessarily
distinct) eigenvalues as discs.
- [A: real] If the discs are all distinct then
the eigenvalues are all real.
- If an eigenvalue has algebraic multiplicity m then it must lie in
at least m discs.
The minimum polynomial, f(t) of a square matrix
A[n#n] is the unique monic polynomial of least
degree for which f(A)=0.
An eignevalue is regular if its geometric
and algebraic multiplicities are equal, otherwise it is
An eigenvalue is simple if its algebraic
multiplicity is equal to 1. This implies that its geometric multiplicity is also equal to 1.
- A simple eigenvalue is always regular.
[m>=n] The singular values of
A[m#n] are the positive square roots of the
- If A[n#n] is normal, its singular values are the absolute values
of its eigenvalues.
- A[n#n] is non-singular iff all its singular
values are > 0.
- The condition number of a matrix is its
largest singular value divided by its smallest singular value.
See also: Singular Value Decomposition
This page is part of The Matrix Reference
Manual. Copyright © 1998-2019 Mike Brookes, Imperial
College, London, UK. See the file gfl.html for copying
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