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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| =
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.
- 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
sum(ABS(A))) <= |c| <=
- 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.
- Similar matrices have the same
- The roots of the minimum and characteristic
polynomials are identical (though their multiplicities may differ) and are the
eigenvalues of A.
- The minimum polynomial of A[n#n] is a
factor of its characteristic polynomial and its order
is <= n.
- If A is nilpotent to index
k, its minimal polynomial is tk.
[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 © 1998-2005 Mike Brookes, Imperial
College, London, UK. See the file gfl.html for copying
instructions. Please send any comments or suggestions to "mike.brookes" at
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