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.
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.
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)
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 $\displaystyle A$ ) .
Use that if $\displaystyle P^{-1}AP=J$ then, $\displaystyle e^A=Pe^JP^{-1}$ .b)Let A is a realnumber matrix nxn. Show that det (exp (A)) = det (exp (J)) where J isthe Jordan canonical form of A.
Said well, support you ！
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