MHB Jacqueline's question at Yahoo Answers (Eigenvalues of a composition)

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The discussion centers on a 3x3 matrix A that represents a 90-degree rotation around the z-axis and a stretch by a factor of 3 along the z-axis. The proposed matrix is confirmed to be A = [[0, -1, 0], [1, 0, 0], [0, 0, 3]]. The eigenvalues identified are ±i and 3, with corresponding eigenspaces derived for each eigenvalue. The eigenspace for the real eigenvalue 3 is spanned by the vector (0, 0, 1), while the complex eigenvalues have associated eigenspaces represented by (i, 1, 0) and its conjugate. The discussion provides clarity on the matrix's properties and confirms the calculations related to eigenvalues and eigenspaces.
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Here is the question:

Denote by A the 3x3 matrix which rotates by 90 degrees around the z-axis and stretches by a factor of 3 a long the z-axis. Find all eigen values and eigenspaces corresponding to the real eigenvalues.
Would it look something like this..?

0 -3 0
1 0 0
0 0 1

I believes I can the eigenvalues and eigenspaces but I just want to check if my matrix is correct first.

Here is a link to the question:

Denote by A the 3x3 matrix which rotates by 90 degrees around the z-axis? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Jacqueline,

The matrix $A$ corresponds to the following composition: $$A=\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}& {0}&{3}\end{bmatrix}\begin{bmatrix}{\cos \pi/2}&{-\sin \pi/2}&{0}\\{\sin \pi/2}&{\;\;\cos \pi/2}&{0}\\{0}&{0}&{1}\end{bmatrix}=\begin{bmatrix}{0}&{-1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{3}\end{bmatrix}$$ The eigenvalues of $A$ are $\pm i$ and $3$. The corresponding eigenspaces: $$\ker (A-3I)\equiv \left \{ \begin{matrix}-3x_1-x_2=0\\x_1-3x_2=0\\0=0\end{matrix}\right.\Rightarrow B_{3}=\{(0,0,1)^T\}\\\ker (A-iI)\equiv \left \{ \begin{matrix}-ix_1-x_2=0\\x_1-ix_2=0\\(3-i)x_3=0\end{matrix}\right.\Rightarrow B_{i}=\{(i,1,0)^T\}$$ As $A$ is a real matrix, $B_{-i}=\{(\bar{i},\bar{1},\bar{0})^T\}=\{(-i,1,0)^T\}$.

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