# Decomposition of a matrix into a skew symmetric matrix and “positive definite” symmetric matrix

I have a special case which is slightly different from the usual decomposition of a matrix $A$ into skew symmetric and symmetric matrices ($A=\frac{1}{2}(A-A^T)$+ $\frac{1}{2}(A+A^T)$).
Here I need to have the symmetric part to be "POSITIVE Definite" as well.

Is there another decomposition technique which guarantees to have the symmetric matrix also positive definite?

• The result should be a sum, a product.. of these matrices, or something else? – Peter Franek Oct 31 '16 at 8:22
• The decomposition of a matrix as a sum of a symmetric and an anti-symmetric is unique, so if you need positive definite, you need some other kind of decomposition. What kind are you looking for? – Gerry Myerson Nov 2 '16 at 8:10
• Thanks. The result should be a sum of a skew symmetric matrix and a positive definite one – Alex Ren Dec 4 '16 at 12:06