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).