- #1
fluidistic
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Homework Statement
Write the A matrix and the x vector into a basis in which A is diagonal.
[itex]A=\begin{pmatrix} 0&-i&0&0&0 \\ i&0&0&0&0 \\ 0&0&3&0&0 \\ 0&0&0&1&-i \\ 0&0&0&i&-1 \end{pmatrix}[/itex].
[itex]x=\begin{pmatrix} 1 \\ a \\ i \\ b \\ -1 \end{pmatrix}[/itex].
Homework Equations
A=P^(-1)A'P.
The Attempt at a Solution
I found out the eigenvalues (spectra in fact) to be [itex]\sigma (A) = \{ -1,0,0,1,3 \}[/itex].
I'm happy they told me A is diagonalizable; so I can avoid finding the Jordan form of A.
So I know how is A'. I think that in order to find P, I must find the eigenvectors associated with each eigenvalues. P would be the matrix whose columns are the eigenvectors encountered. So with [itex]\lambda = -1[/itex], I found [itex]v_1=\begin{pmatrix} -i \\ i \\ 3 \\ 4-i \\ i+1 \end{pmatrix}[/itex]. But for [itex]\lambda =0[/itex] I reach the null vector as eigenvector, which I know is impossible.
Am I doing something wrong?