Matrix congruence

In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that

PTAP = B

where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation.

Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases.

Note that Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose,[1] but this definition has not been adopted by most other authors.

Congruence over the reals

Sylvester's law of inertia states that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.[2]

gollark: This is very troubling. Even in release mode, the nim markdown parser is about a thousand times slower than the rust one.
gollark: I wonder if anyone tried making some cool lisp-styled assembler so you could have more unified macros.
gollark: Frankly, I'm tempted to just make minoteaur support regularized HTML or some BBCode derivative.
gollark: Link to this?
gollark: Markdown cheatsheets are also not usable as a Markdown spec. Markdown does not actually *have* a spec, so we have a wild west of incompatible implementations. Some try to mimic the original perl script, some just do approximately the right thing in most cases, some do the easy thing in case of weirdness, some follow one of many subtly incompatible formal specs.

See also

References
  1. Halmos, Paul R. (1958). Finite dimensional vector spaces. van Nostrand. p. 134.
  2. Sylvester, J J (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares" (PDF). Philosophical Magazine. IV: 138–142. Retrieved 2007-12-30.
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