Show that, if is any complex-valued matrix,

where is the trace.

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- Aug 18th 2009, 04:22 PMBruno J.An identity of Jacobi
Show that, if is any complex-valued matrix,

where is the trace.

- Aug 18th 2009, 04:38 PMynj
- Aug 18th 2009, 04:40 PMBruno J.
is a matrix with complex entries.

- Aug 18th 2009, 04:46 PMynj
- Aug 18th 2009, 05:12 PMBruno J.
- Aug 18th 2009, 05:56 PMynj
Let be a polynomial,if the eigenvalue of is , then the eigenvalue of is

we may look as a polynomial series of (limit)

note that:

but is the eigenvalue of (regard it as polynomial)

so - Aug 18th 2009, 06:20 PMBruno J.
Remember that not every matrix is diagonalizable!

In the case where the matrix is diagonal then it is easy to show that the identity holds because if is one of the diagonal entries, the corresponding entry of will be . - Aug 18th 2009, 06:53 PMynj
I have changed my proof

every matrix have eigenvalue, and my proof is not based on diagonalibility - Aug 18th 2009, 11:09 PMCaptainBlack
See here (2/3rd of the way down)

CB