Matrix congruence
Appearance
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
eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.[2]
See also
References
- van Nostrand. p. 134.
- ^ 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.
- Gruenberg, K.W.; Weir, A.J. (1967). Linear geometry. van Nostrand. p. 80.
- Hadley, G. (1961). Linear algebra. Addison-Wesley. p. 253.
- ISBN 0-471-02371-X.
- ISBN 0-486-66434-1.
- ISBN 0-486-67102-X.
- Norman, C.W. (1986). Undergraduate algebra. ISBN 0-19-853248-2.