# Is a product of a Hurwitz matrix and a diagonal positive definite matrix always Hurwitz?

If I multiply a Hurwitz matrix (real part of eigenvalues are negative) with a diagonal positive definite matrix, does the product still remain as Hurwitz matrix?

• What is your definition of a Hurwitz matrix? – Arman Malekzadeh Jun 18 '17 at 6:55
• A matrix is said to be Hurwitz iff its eigenvalues have a strictly negative real part. – palas Jun 18 '17 at 7:27

How about $$A=\pmatrix{1&1\\-4&-3}$$ and $$B=\pmatrix{4&0\\0&1}.$$ Then $A$ has repeated eigenvalues $-1$ so is Hurwitz, but $BA$ has positive trace so isn't.