Matrix equivalence

In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if

[math]\displaystyle{ B = Q^{-1} A P }[/math]

for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.

The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same endomorphism V → V under two different choices of a single basis of V, used both for initial vectors and their images.

Properties

Matrix equivalence is an equivalence relation on the space of rectangular matrices.

For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions

The matrices can be transformed into one another by a combination of elementary row and column operations.
Two matrices are equivalent if and only if they have the same rank.

Canonical form

The rank property yields an intuitive canonical form for matrices of the equivalence class of rank [math]\displaystyle{ k }[/math] as

[math]\displaystyle{ \begin{pmatrix} 1 & 0 & 0 & & \cdots & & 0 \\ 0 & 1 & 0 & & \cdots & & 0 \\ 0 & 0 & \ddots & & & & 0\\ \vdots & & & 1 & & & \vdots \\ & & & & 0 & & \\ & & & & & \ddots & \\ 0 & & & \cdots & & & 0 \end{pmatrix} }[/math],

where the number of [math]\displaystyle{ 1 }[/math]s on the diagonal is equal to [math]\displaystyle{ k }[/math]. This is a special case of the Smith normal form, which generalizes this concept on vector spaces to free modules over principal ideal domains.

v t e Matrix classes
Explicitly constrained entries	(0,1) Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Binary Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Markov Metzler Monomial Moore Nonnegative Partitioned Parisi Pentadiagonal Permutation Persymmetric Polynomial Positive Quaternionic Sign Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Unitary Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Diagonalizable Hurwitz Positive-definite Stability Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Orthonormal Singular Unimodular Unipotent Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Derogatory Distance Duplication Elimination Euclidean distance Fundamental (linear differential equation) Generator Gramian Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation Wedderburn X–Y–Z
Used in statistics	Bernoulli Centering Correlation Covariance Design Dispersion Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Skew-adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian
List of matrices Category:Matrices

Matrix equivalence

Properties

Canonical form

See also