Congruence (matrix)

In linear algebra , a branch of mathematics , one calls two square matrices and congruent if there is an invertible matrix such that: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle P}$

${\displaystyle B=P^{T}AP}$.

The matrix to be transposed means . The congruence of matrices is an equivalence relation on the class of the quadratic matrices . ${\displaystyle P^{T}}$ ${\displaystyle P}$

Equivalently, it can be defined that two matrices are congruent if they represent the same bilinear form with respect to two (possibly different) bases .

According to Sylvester's law of inertia , two real symmetric matrices are congruent if and only if they have the same index of inertia . The inertia index is the triple consisting of the number of positive, negative and zero eigenvalues .

See also

• Equivalence (matrix)
• Similarity (matrix)

literature

• Michael Artin : Algebra. Birkhäuser, Basel et al. 1998, ISBN 3-7643-5938-2 , Chapter 8: Linear groups .