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 Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

P^TAP = B

where "T" denotes the matrix transpose. Sylvester's law of inertia states that two congruent symmetric real matrices have the same numbers of positive and negative eigenvalues. In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ... In linear algebra, Sylvesters law of inertia states that the inertia of a matrix A is invariant under congruence transformations. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...