Matrix congruence
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In mathematics, two matrices A and B with real entries are called congruent if there exists an invertible matrix P with real entries such that
- PTAP = B
where "T" denotes the matrix transpose.
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.
Sylvester's law of inertia states that two congruent symmetric real matrices 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.[1]
[edit] References
- ^ 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". Philosophical Magazine IV: 138–142.
