Eigenvalues

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The eigenvalues of a square 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 an algebraic multiplicity (or just multiplicity) of k iff (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.
• If, for each of the eigenvalues, the algebraic multiplicity equals the geometric multiplicity, then the matrix is diagonalizable, otherwise it is defective.

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|

Defective Eigenvalue

An eigenvalue is defective if its geometric multiplicity is less than its algebraic multiplicity.

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.
• [Normal]: The singular values of A equal the absolute values of the eigenvalues.
• [Hermitian +ve semidefinite]: The singular values of A equal the eigenvalues.
• 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.

Eigenvector

• The eigenvectors associated with an eigenvalue, c, form a subspace whose dimension equals the geometric multiplicity of c and never exceeds the algebraic multiplicity of c. This subspace is called the eigenspace corresponding to c.
• A matrix A_n#n has an orthonormal set of n eigenvectors iff it is normal. In this case we may write A=UDU^H for a diagonal matrix, D, of eigenvalues and a unitary matrix, U, whose columns are the corresponding eigenvectors.
• If  Ax = cx. and  A^Hy = dy where c!=d^*, then y^Hx = 0, i.e. x and y are orthogonal.
  • [A:Hermitian]: Eigenvectors corresponding to distinct eigenvalues of A are orthogonal.
• If B and A are similar with B=X^-1AX for some non-singular X, then y is an eignvector of B iff Xy is an eigenvector of A corresponding to the same eigenvalue.

Geometric Multiplicity

• The geometric multiplicity of an eigenvalue is less than or equal to its algebraic multiplicity.
• If, for each of the eigenvalues, the algebraic multiplicity equals the geometric multiplicity, then the matrix is diagonalizable, otherwise it is defective.

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; Gersgorin Discs make this statement precise.

• 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.
• The minimum polynomial of a matrix is equal to its characteristic polynomial iff each eigenvalue has a geometric multiplicity of 1.
• If A is nilpotent to index k, its minimal polynomial is t^k.

Regular Eigenvalue

An eignevalue is regular if its geometric and algebraic multiplicities are equal, otherwise it is defective.

Simple Eigenvalue

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.

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