How To Compute The Signature
There are some methods for computing the signature of a matrix.
- For any nondegenerate symmetric matrix of n × n, diagonalize it (or find all of eigenvalues of it) and count the number of positive and negative signs, and get p and q = n − p, they may take a pair of values from 0 to n, then the signature will be s = p − q.
- The sign of the roots of the characteristic polynomial may be determined by Cartesius' sign rule as long as all roots are reals.
- Lagrange algorithm gives a way to compute an orthogonal basis, and thus compute a diagonal matrix congruent (thus, with the same signature) to the other one: the signature of a diagonal matrix is the number of positive, negative and zero elements on its diagonal.
- According to Jacobi's criterion, a symmetric matrix is positive-definite if and only if all the determinants of its main minors are positive.
Read more about this topic: Metric Signature
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