Matrix Reference Manual
Matrix Decompositions


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Cholesky Decomposition

Iff A is hermitian positive definite there exists a non-singular upper triangular U with UHU=A . This is the Cholesky decomposition of A.


Dual Conjunctive Diagonalization

[A,B:n#n, hermitian] If A is positive definite there exists X such that XHAX=I and XHBX=DIAG(k) where k contains the eigenvalues of A-1B.


Hermite Normal Form or Row-Echelon Form

Every matrix A:m#n can be expressed as A=BCP where B:m#m is non-singular, P:n#n is a permutation matrix and C:m#n is of the form [I D;0] for some D. The matrix C is the row-echelon or Hermite-normal form of A.


Jordan Normal Form

For any A:n#n, there exists X such that A=X-1BX where B is of the following form:

  1. B is upper bidiagonal and its main diagonal consists of the eigenvalues of A repeated according to their algebraic multiplicities.
  2. B = DIAG(X,Y,…) where each square block X, Y, … has one of A's eigenvalues repeated along its main diagonal, 1's along the diagonal above it and 0's elsewhere (these blocks are hypercompanion matrices).

Finding the Jordan form of a defective or nearly defective matrix is numerically illconditioned.


LDU Decomposition

Every non-singular square matrix A can be expressed as A=PLDU where P is a permutation matrix, L is unit lower triangular, D is diagonal and U is unit upper triangular.


LU Decomposition

Every square matrix A can be expressed as A=PLU where P is a permutation matrix, L is a unit lower triangular matrix and U is upper triangular.

This decomposition is the standard way of solving the simultaneous equations Ax = b.


Polar Factorisation

Every A[n#n] can be expressed as A=UH=KV with U, V unitary and H, K positive semidefinite hermitian. H and K are unique; U and V are unique iff A is nonsingular. If A is real then H and K are real and U and V may be taken to be real.


QR Decomposition

For any A:m*n with m>=n, A=QR for Q:m*m unitary and R:m*n upper triangular.


Schur Decomposition

Every square matrix A is unitarily similar to an upper triangular matrix T: A=UHTU.


Schur Decomposition, Real

Every square real matrix A is orthogonally similar to an upper block triangular matrix T: A=QTTQ where each block of T is either a 1#1 matrix or a 2#2 matrix having complex conjugate eigenvalues.


Schur Decomposition, Generalized

If A and B are square matrices, then there exist unitary U and V and upper triangular S and T such that A=UHSV and A=UHTV.


Schur Decomposition, Generalized Real

If A and B are real square matrices, then there exist orthogonal U and V and upper block triangular S and T such that A=UTSV and A=UTTV  where each block of S and T is either a 1#1 matrix or a 2#2 matrix having complex conjugate eigenvalues.


Singular Value Decomposition (SVD)

Every matrix A:m#n can be expressed as A=UDVH where U:m#m and Vn#n are unitary and D:m#n is diagonal with real positive diagonal elements arranged in non-increasing order (i.e. if j>i then d(j,j) >= d(i,i)). The diagonal elements of D are the singular values of A.