Matrix Relations

Congruence

Square matrices A and B are congruent if there exists a non-singular X such that B= X^TAX .

For Hermitian congruence, seer Conjuctivity.

Congruence implies equivalence.

• Congruence is an equivalence relation.
• Congruence preserves symmetry, skewsymmetry and definiteness
• A is congruent to a diagonal matrix iff it is Hermitian.

Conjunctivity

Square matrices A and B are conjunctive or hermitely congruent if there exists a non-singular X such that B= X^HAX.

• If A is hermitian, it is conjunctive to a matrix of the form DIAG(I,-I,0).
• If A is skew-hermitian, it is conjunctive to a matrix of the form DIAG(jI,-jI,0).
• A is conjunctive to I iff it is positive definite hermitian in which case A=U^HU for some non-singular upper triangular U.

Equivalence

Two m*n matrices, A and B, are equivalent iff there exists a non-singular m*m matix Mand a non-singular n*n matrix N with B=MAN .

• Equivalence is an equivalence relation.
• A and B, are equivalent iff they have the same rank.

Kronecker Product

The Kronecker product of A:m#n and B:p#q is equal to the mp#nq matrix [a(1,1)B … a(1,n)B ; … ; a(m,1)B … a(m,n)B ]. It is also known as the direct product or tensor product of A and B. The Kronecker Product operation is denoted by a × sign enclosed in a circle.

• Kronecker products are associative and distributive over matrix addition:
  • KRON(A,B,C) = KRON(A,KRON(B,C)) = KRON(KRON(A,B),C)
  • KRON(A,B+C) = KRON(A,B) + KRON(A,C)
  • KRON(A^T,B^T) = (KRON(A,B))^T
• KRON(A^H,B^H) = (KRON(A,B))^H
• KRON(A^-1,B^-1) = (KRON(A,B))^-1
  • KRON(A,B) is singular iff A or B is singular.
• KRON(AB,CD) = KRON(A,C)KRON(B,D)
  • (KRON(A,B))^HKRON(A,B) = KRON(A^HA,B^HB)
• [A,B: n#n] tr(KRON(A,B)) = tr(A) tr(B)
• [A:n#n, B:m#m] The mn eigenvalues of KRON(A,B) are equal to the mn possible products of an eigenvalue of A multiplied by an eigenvalue of B with the appropriate algebraic multiplicities.

By converting an unknown matrix X into a long column vector x(:) by concatenating the columns of X, the following property allows the solution of matrix equations that involve sums of the form AXB+CXD+EXF +… = Z. This is however a slow and often ill-conditioned way of solving such equations.

• If Y=AXB+CXD then y(:) = (KRON(B^T,A) + KRON(D^T,C)) x(:)
  • If Y=AX then y(:) = KRON(I,A) x(:)

Orthogonal Similarity

Real square matrices A and B are orthogonally similar if there exists an orthogonal Q such that B= Q^TAQ .

Orthogonal similarity implies both similarity and congruence.

See also: Unitary similarity

Similarity

Square matrices A and B are similar if there exists a non-singular X such that B=X^-1AX .

Similar matrices represent the same linear transformation in a different basis. Similarity implies equivalence .

• Similar matrices have the same trace, determinant, eigenvalues and characteristic equation.

Unitary Similarity

Square matrices A and B are unitarily similar if there exists a unitary Q such that B= Q^HAQ .

Unitary similarity implies both similarity and conjunctivity. If A and B are real, they are orthogonally similar .

The Matrix Reference Manual is written by Mike Brookes, Imperial College, London, UK. Please send any comments or suggestions to mike.brookes@ic.ac.uk