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I am interested to find out the proof for the following statement (it's from my textbook and it is stated without proof):

A symmetric matrix is negative definite if and only if all of its principal minors of even order are positive and all of its principal minors of odd order are negative.

Based on the proof for the case of positive definite symetric matrix on Wikipedia, I think this proof will be more complicated. So here are the specific things that I hope someone can elaborate on:

  1. What are the theorems I need to be aware of to understand this proof?
  2. Why does the sign of the odd and even order of the principal minors matter in the case of the negative symmetric matrix while in the case of the positive symmetric matrix, all of its principal minors must be positive?

Thank you.

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Sketch: Assuming known/proved the analogus property for positive definite matrices, and knowing that ${\mathbf A}$ is negative definite if and only if $-{\mathbf A}$ is positive definite, think about the principal minors of $-{\mathbf A}$. These are determinants of the negated submatrices, and the property $|c {\mathbf A}| = c^n |{\mathbf A}|$ tells us that they will be equal to the original determinants if the size of the submatrix (order of the principal minor) is even, or its negated value if odd. Etc.

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