Understanding the Relationship: Log, Traces, and Diagonalized Matrices

[dm4b]

Trying to make sense of the following relation:

\sum log d_{j} = tr log(D)

with D being a diagonalized matrix.

Seems to imply the log of a diagonal matrix is the log of each element along the diagonal.

Having a hard time convincing myself that is true, though

 

[dm4b]

One more:

if M = A^{-1}DA,

why is this true:

tr A^{-1}log(D)A=tr\ log (M)

 

[Office_Shredder]

Trying to make sense of the following relation:

\sum log d_{j} = tr log(D)

with D being a diagonalized matrix.

Seems to imply the log of a diagonal matrix is the log of each element along the diagonal.

Having a hard time convincing myself that is true, though

No, this is a fact about eigenvalues. If \lambda is an eigenvalue of D, then \log(\lambda) is an eigenvalue of \log(D).

This is usually first presented in the other direction, that if \lambda is an eigenvalue of D, then e^{\lambda} is an eigenvalue of e^D. (and the eigenvector is the same). It's very easy to see this by writing out the power series definition of e^D and applying it to the eigenvector of D

 

[dm4b]

No, this is a fact about eigenvalues. If \lambda is an eigenvalue of D, then \log(\lambda) is an eigenvalue of \log(D).

This is usually first presented in the other direction, that if \lambda is an eigenvalue of D, then e^{\lambda} is an eigenvalue of e^D. (and the eigenvector is the same). It's very easy to see this by writing out the power series definition of e^D and applying it to the eigenvector of D

ahh, thanks I should have known that.

I figured out the second one too, so no help needed on that one now.