Eigenvalues

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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].

Algebraic Multiplicity

An eigenvalue c has algebraic multiplicity k if (t-c)^k is the highest power of (t-c) that divides the characteristic polynomial.

• The algebraic multiplicity of an eigenvalue is greater than or equal to its geometric multiplicity.

Characteristic Equation

[n*n] The characteristic equation of a matrix A is |tI-A| = 0. It is a polynomial equation in t.

• [n*n] A matrix A satisfies its own characteristic equation (Cayley-Hamilton theorem)

Characteristic Matrix

[n*n]: The characteristic matrix of A is (tI-A) and is a function of the scalar t.

Characteristic Polynomial

[n*n] The characteristic polynomial, p(t), of a matrix A is p(t) = |tI - A|.

• [n*n]: The characteristic polynomial of A is of the form: tⁿ - tr(A)*t^n-1 + ... + -1ⁿ |A|.
  • [2*2]: |tI-A| = t² - tr(A)*t + |A|
• [A,B: m*n]: If m>n |tI - AB'| = t^m-n * |tI - B'A|
• [n*n]: |tI-AB| = |tI-BA|

Eigenvalues

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 t.
• [Complex, n*n]: The matrix A has exactly n eigenvalues (not necessarily distinct)
• [Complex]: tr(A) = sum(eig(A))
• [Complex]: det(A) = prod(eig(A))
• [A:m*m, C:n*n]: eig([A B; 0 C]) = [eig(A); eig(C)]
• det(A)=0 iff 0 is an eigenvalue of A
• The eigenvalues of a triangular or diagonal matrix are its diagonal elements.
• [Hermitian]: The eigenvalues of A are all real.
• [Unitary]: The eigenvalues of A have unit modulus.
• [Nilpotent]: The eigenvalues of A are all zero.
• [Idempotent]: The eigenvalues of A are all either 0 or 1.
• The eigenvalues of A^k are (eig(A))^k
• Similar matrices have the same eigenvalues
• [Real, Symmetric, 2#2] A=[a b; b d]=RDR^T 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.

Geometric Multiplicity

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 geometric multiplicity of an eigenvalue is less than or equal to its algebraic multiplicity.

Gersgorin Discs

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 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.

Minimum Polynomial

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 minimum polynomial.
• 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 t^k.

Singular Values

[m>=n] The singular values of A_[m#n] are the positive square roots of the eigenvalues of A^HA.

• 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