Inner Product
For an m-by-n matrix A with complex (or real) entries and * being the conjugate transpose, we have
with equality if and only if A = 0. The assignment
yields an inner product on the space of all complex (or real) m-by-n matrices.
The norm induced by the above inner product is called the Frobenius norm. Indeed it is simply the Euclidean norm if the matrix is considered as a vector of length mn.
It follows that if A and B are positive semi-definite matrices of the same size then
Read more about this topic: Trace (linear Algebra)
Famous quotes containing the word product:
“Perhaps I am still very much of an American. That is to say, naïve, optimistic, gullible.... In the eyes of a European, what am I but an American to the core, an American who exposes his Americanism like a sore. Like it or not, I am a product of this land of plenty, a believer in superabundance, a believer in miracles.”
—Henry Miller (1891–1980)
“Out of the thousand writers huffing and puffing through movieland there are scarcely fifty men and women of wit or talent. The rest of the fraternity is deadwood. Yet, in a curious way, there is not much difference between the product of a good writer and a bad one. They both have to toe the same mark.”
—Ben Hecht (1893–1964)