Can anyone help me showing that

a)Let A is a realnumber matrix nxn. Show that tr (A) =sum of all eigenvalues of A.

b)Let A is a realnumber matrix nxn. Show that det (exp (A)) = det (exp (J)) where J is

the Jordan canonical form of A.

Printable View

- Dec 12th 2011, 08:54 AMmathsohardtrace, jordan carnonical form
Can anyone help me showing that

a)Let A is a realnumber matrix nxn. Show that tr (A) =sum of all eigenvalues of A.

b)Let A is a realnumber matrix nxn. Show that det (exp (A)) = det (exp (J)) where J is

the Jordan canonical form of A. - Dec 12th 2011, 09:36 AMSironRe: trace, jordan carnonical form
We know that $\displaystyle \mbox{Tr(A)}=\sum_{i=1}^{n}a_{ii}=a_{11}+a_{22}+.. .a_{nn}$

Now, have you any idea about the matrix of the linear map? (I would choose a base wherefore the matrix of the linear map is a diagonal matrix) - Dec 12th 2011, 09:47 AMmathsohardRe: trace, jordan carnonical form
I know TAT^-1 is D and D is the diagonal matrix whose diagonal entries are the eigenvalues of A. Hmm. I think I kind of getting it but I think I need a smoother answer please!

- Dec 12th 2011, 01:59 PMFernandoRevillaRe: trace, jordan carnonical form
**Siron**'s answer is correct for diagonalizable matrices in $\displaystyle \mathbb{R}$ . In general:

Use that similar matrices have the same trace and $\displaystyle A$ is similar to its canonical form of Jordan (triangular matrix whose elements in the diagonal are__exactly__the eigenvalues of $\displaystyle A$ ) .

Quote:

b)Let A is a realnumber matrix nxn. Show that det (exp (A)) = det (exp (J)) where J isthe Jordan canonical form of A.

- Dec 13th 2011, 09:02 PMmdiendd123456Re: trace, jordan carnonical form
Said well, support you ！

___________________

**Schaufensterpuppen**