[Math] Symmetric Matrix as the Difference of Two Positive Definite Symmetric Matrices

linear algebramatrices

Prove that any real symmetric matrix can be expressed as the difference of two
positive definite symmetric matrices.

I was trying to use the fact that real symmetric matrices are diagonalisable , but the confusion I am having is that 'if $A$ be invertible and $B$ be a positive definite diagonal matrix, then is $ABA^{-1}$ positive definite' .

Thanks for any help .

Best Answer

Let $S$ be your symmetric matrix. You can now add a large positive multiple of the identity matrix. This ensures that your matrix $S+c I$ is diagonally dominant and symmetric, and thus positive definite.

See

http://mathworld.wolfram.com/DiagonallyDominantMatrix.html

Now, you clearly have $S= (S+cI)-cI$ (and $c I$ is certainly positive definite as well).

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