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petergreat
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Is every complex symmetric (NOT unitary) matrix [tex]M[/tex] diagonalizable in the form [tex]U^T M U[/tex], where [tex]U[/tex] is a unitary matrix? Why?
g_edgar said:Woah, [tex]U^T[/tex] in your formula and not [tex]U^*[/tex] ... so in general [tex]U^T[/tex] is not the inverse of [tex]U[/tex] . Why did you choose that?
petergreat said:Is every complex symmetric (NOT unitary) matrix [tex]M[/tex] diagonalizable in the form [tex]U^T M U[/tex], where [tex]U[/tex] is a unitary matrix? Why?
Yes, that was a typo.jostpuur said:You probably meant to emphasize "symmetric (not hermitian)"?
jostpuur said:My belief is that if [itex]M\in\mathbb{C}^{n\times n}[/itex] is symmetric so that [itex]M^T=M[/itex], then there exists a complex orthogonal matrix [itex]O\in\mathbb{C}^{n\times n}[/itex] so that [itex]O^T=O^{-1}[/itex], and so that [itex]O^TMO[/itex] is diagonal. (And I believe that the answer to your question is: No.)
Unfortunately I don't have a reference for this claim, and I also don't have energy to go through a proof right now, because this isn't my problem, so you shouldn't believe my belief
jostpuur said:Can you show explicitly your example matrices that you have been working on?
Diagonalizing a complex symmetric matrix means transforming it into a diagonal matrix, where all the non-diagonal elements are zero. This simplifies the matrix and makes it easier to perform calculations and solve equations involving the matrix.
A complex symmetric matrix can be diagonalized by finding its eigenvalues and eigenvectors. The eigenvectors form a basis for the matrix and can be used to transform it into a diagonal matrix.
Yes, all complex symmetric matrices can be diagonalized. This is because they have a full set of eigenvectors, which are necessary for diagonalization.
Diagonalizing a complex symmetric matrix can make it easier to solve equations involving the matrix, as well as make it easier to identify patterns and relationships within the matrix. It can also simplify the matrix and make it more computationally efficient to work with.
Yes, diagonalization of complex symmetric matrices is commonly used in various fields of science and engineering, such as physics, chemistry, and electrical engineering. It is also used in computer algorithms for tasks such as data compression and image processing.