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Let $C\in Mn$ symmetric matrices which have k positive diagonal entries, and n-k negative diagonal entries, where is $k\in \{1,2...,n-1\}$. Try to figure when that matrices positive definition, when is negative definition, and when is not definite. Explain the answer.

If matrices positive definite, the pivots must be positive, here first pivot is $a11$ so it must be positive,so we have $k\geq1$ but i know that diagonal entries must be positive if you want positive definite matrices, because if you have $ei^{T}Aei=0\cdot a11\cdot0+....1\cdot aii\cdot 1$ and your diagonal entries are not positive than matrices do not positive definite for every eigenvector,but here we can only have n-1 positive diagonal entries and i still have one negative diagonal entries, and how it can be positive definite?

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Guessing That you are in R, using Sylvester theorem, you know That every symmetric bilinear form (or quadratic form) has a base in which the form is a diagonal form with k elements in the trace that are positive and r-k elements of the trace that are negative, where r is the rank and (r-k) is defined ad negativity index. But k belongs to {0,1,2,....r} So the theorem states That the positive definite, negative definite, semi-positive definite, semi-negative definite and indefinite matrices form a complete partition od symmetric matrices in R; This means That every symmetric matrix you have is similar to One of those I mentioned before. In particular if r=n=k then It is positive definite, if r=n and k=0 then It is negative definite, if r

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