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Two real symmetric matrices $A$ and $B$ are called congruent if there exists an invertible matrix $P$ such that $$P^TAP=B$$ I am aware that the number of positive, negative, and zero eigenvalues is an invariant under congruence. Are there any other quantities that are invariant under congruence?

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Sylvester's law of inertia states that two symmetric matrices are congruent if and only if they have the same number of positive, negative, and zero eigenvalues (i.e. the same inertia). In other words, there is nothing in addition to the inertia that is preserved by congruence.

There are quantities that can be written as a function of the inertia of a matrix; these quantities must be preserved by matrix congruence. For example, the sign of the determinant and the rank must be the same for congruent matrices $A$ and $B$. Whether you consider these to be "other" invariant quantities depends on what you have in mind.

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  • $\begingroup$ Note that the "if" part (which is what you need to conclude that nothing else is preserved) is pretty easy, especially once you know that symmetric forms can be diagonalised. The "only if" part is the more difficult one. $\endgroup$
    – tomasz
    Jul 14, 2021 at 9:45

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