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

  • Thread starter Thread starter Fernando Revilla
  • Start date Start date
  • Tags Tags
    Composition
AI Thread Summary
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.
Fernando Revilla
Gold Member
MHB
Messages
631
Reaction score
0
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.
 
Mathematics news on Phys.org
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 http://www.mathhelpboards.com/f14/ section.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top