We have three square matrices A, B, C of the same dimension. Moreover B and C are positive semidefinite. In my analysis I need to calculate Tr[A.(B*C)], where * denotes the Hadamard product and . the standard matrix product. For some reason (I don't want to go into details) this representation is not so convenient for me. Do you know any kind of relation between the Hadamard product and standard matrix product so that this formula could be transformed?
J.R. Schott, Matrix Analysis for Statistics, Second Edition,
John Wiley & Sons, Inc., New Jersey, 2005.
Theorem 7.28. Let A and B be m×m nonnegative definite matrices (with ordered eigenvalues λ₁≥…≥ λ_{m}). Then
λ_{m}(A∘B) ≥ λ_{m}(AB).
On p.315
Problem 35. Let A, B, and C he m×n matrices. Show that
tr{(A′∘B′)C} = tr{A′(B∘C)}.
It seems unlikely, a priori, that there is some nice way of rearranging things --- the trace exhibits *some* nice behaviour (invariance under cyclic rotations of products) but I doubt that there will be a nice way to get it to interact with the Hadamard product. Perhaps you could be more explicit about what things are difficult for you to compute (usually Hadamard product is very easy!) and what sort of answer would be helpful (even if it isn't one that works, it would give an idea of what you are looking for).
Not sure if this is any help but you can rewrite Trace(A^T.(B*C)) as a^T.diag(b).c, where a, b, and c are the vectorized versions of A, B, and C respectively.
Dir sir,
In general no, but inequalities exist on trace of hadamard products for positive semi-definite matrices.
J.R. Schott, Matrix Analysis for Statistics, Second Edition,
John Wiley & Sons, Inc., New Jersey, 2005.
Theorem 7.28. Let A and B be m×m nonnegative definite matrices (with ordered eigenvalues λ₁≥…≥ λ_{m}). Then
λ_{m}(A∘B) ≥ λ_{m}(AB).
On p.315
Problem 35. Let A, B, and C he m×n matrices. Show that
tr{(A′∘B′)C} = tr{A′(B∘C)}.
About efficient computation: for any two matrices X and Y, Trace (X^T.Y)= x^Ty, where x and y denote the vectorized versions of X and Y ( the expression actually is the scalar (inner) product of matrices). In your case X=A^T, Y=B*C.
Dear Wiwat Wanicharpichat!
Thank you very much for your contribution. The relation from Problem 35. is exactly the one that allows me to transform my formulas into the desired form. Thakns to your help I can continue working on my new paper :)
I appreciate also all the other responses to my question. Thanks for devoting your time to help me :)
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