The Hadamard product or pairwise product of two matrices denoted hereby by "o"
C = A o B.
Let Lambda(A) be the set of all eigenvalues of matrix A.
I am aware of inequalities relating Lambda(A), Lambda(B), and Lambda(C).
It is possible that my question does not have any answer but I would just like to make sure that is there any closed form EQUALITY expressions relating Lambda(A), Lambda(B), and Lambda(C).
Thank you in advance.
You mean expression of eigenvalues of the product in terms of eigenvalues of the multipliers? There is no such a relation.
Yes i mean so, and yes that's what i thought, i only wanted to make sure that it doesn't exist.
Roughly speaking, only loose inequalities exist on such relations.
In general, product of all eigenvalues of a Hadamard product of two matrices is greater than equal to the product of all eigenvalues of multiplier matrices.
What about if A and B are nonnegative (entrywise) and let us restrict ourselves to those (non -negative) eigenvalues that admit nonnegative eigenvectors (distinguished eigenvalues). Can we say something more by using e.g. Frobenius normal form?
I never spotted any compact formula for such a relationship. I consulted several reliable sources to no avail and seriously doubt whether such an equality relation exists at all (otherwise some wise people would know it). I am referring to Topics in Matrix Analysis, by R.A. Horn and C.R Johnson (Cambridge University Press), and Matrix Theory: Basic Results and Techniques, by F. Zhang (Springer). Please check the literature therein.
Hi Aljosa,
Yes we can. But, unfortunately, loose inequalities exist on that.
I would refer you to:
"Hadamard Products and Multivariate Statistical Analysis", Georges Stayan
Dear Roman,
Actually, we share the same references. I am almost 99% sure that no close form equalities exist, as you stated.
There is a connection. See my paper "Centraliser codes" and the reference to Robson PhD inside.
Dear Patrick, could you perhaps be a bit more precise? Does one consider the Hadamard product as a minor of the Kronecker pr.?
Dear Aljosa,
A hadamard product could be seen as a submatrix of a Kronecker product
There has been an explicit mathematical proof above that such a closed-form expression does not exist. There is not need to be "almost sure" or look for references to convince youself. That's the beauty of mathematics :)
Dear Pavel, I agree about the closed form. However, it is known that e.g. in the case of elementwise non-negative matrices there are nice inequalities between the spectral radius of the Hadamard product and the given matrices, see e.g.
K.M.R. Audenaert, Spectral radius of Hadamard product versus conventional product for non-negative matrices,{\it Linear Algebra Appl.} 432(1) (2010),
R.A. Horn and F. Zhang, Bounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices,{\it Electron. J. Linear Algebra} 20, (2010), 90--94.
A. Peperko, Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequencespaces, {\it Linear Algebra Appl.} 437 (2012), 189--201.
So eventhough the answer to the original question of Ahmad is clear, there are still several closely related questions which are still very much alive and kicking. :)
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