What is the signature of a matrix?

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Thank you very much!

Some webpages say that the signature of a symmetric real matrix is an integer which equals to the number of positive eigenvalues minus the number of negative ones.

However, I am confused by one problem (Problem 4.1.23, Sp81 on Berkeley Problems in Mathematics):

The set of real $3 \times 3$ symmetric matrices is a real, finite-dimensional vector space isomorphic to $\mathbb{R}^6$. Show that the subset of such matrices of signature $(2,1)$ is an open connected subspace in the usual topology on $\mathbb{R}^6$.

So, what are matrices "of signature $(2,1)$", and what is the signature of a matrix?

Many thanks!

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1 Answer

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The signature of a symmetric matrix expresses the number of positive and negative eigenvalues, counting multiplicities. A matrix of size $3$ has signature $(2,1)$ if two of its eigenvalues are positive and one is negative. Equivalently, the signature can be defined in terms of positive and negative entries of the diagonalized matrix.

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