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

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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, $B_{-i}=\{(\bar{i},\bar{1},\bar{0})^T\}=\{(-i,1,0)^T\}$.

If you have further questions, you can post them in the Linear and Abstract Algebra section.