# set of all eigen value of square matrix

• October 18th 2009, 02:16 AM
GTK X Hunter
set of all eigen value of square matrix
Given any square matrix $A$. Let $\sigma(A)$ be the set of all eigen value of $A$, then $\sigma\left(\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] \right) = ...$
• October 18th 2009, 11:54 AM
Opalg
Quote:

Originally Posted by GTK X Hunter
Given any square matrix $A$. Let $\sigma(A)$ be the set of all eigen value of $A$, then $\sigma\left(\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] \right) = ...$

Hint: If $Ax = \lambda x$ then what are $\begin{bmatrix} A & A \\ A & A \end{bmatrix}\begin{bmatrix} x \\ x \end{bmatrix}$ and $\begin{bmatrix} A & A \\ A & A \end{bmatrix}\begin{bmatrix} x \\ -x \end{bmatrix}$ ?
• March 27th 2010, 01:18 AM
GTK X Hunter
what i do is assuming $A = \left[\begin{matrix}1&2\\5&4\end{matrix}\right] \ni \lambda_1=-1,\lambda_2=6$ then form a matrix $B=\left[\begin{matrix} A & A \\ A & A \end{matrix} \right]$ then try to find the eigen values. But it's not easy >_<
• March 27th 2010, 10:35 AM
Opalg
Quote:

Originally Posted by GTK X Hunter
what i do is assuming $A = \left[\begin{matrix}1&2\\5&4\end{matrix}\right] \ni \lambda_1=-1,\lambda_2=6$ then form a matrix $B=\left[\begin{matrix} A & A \\ A & A \end{matrix} \right]$ then try to find the eigen values. But it's not easy >_<

The eigenvectors of A are $\begin{bmatrix}1\\-1\end{bmatrix}$ (for the eigenvalue –1) and $\begin{bmatrix}2\\5\end{bmatrix}$ (for the eigenvalue 6). According to my previous comment, the eigenvectors for B are $\begin{bmatrix}1\\-1\\1\\-1\end{bmatrix},\ \begin{bmatrix}1\\-1\\-1\\1\end{bmatrix},\ \begin{bmatrix}2\\5\\2\\5\end{bmatrix},\ \begin{bmatrix}2\\5\\-2\\-5\end{bmatrix}$. From that, you can check that the eigenvalues are 0 (double eigenvalue), –1 and 6.
• March 27th 2010, 12:14 PM
HallsofIvy
Quote:

Originally Posted by GTK X Hunter
what i do is assuming $A = \left[\begin{matrix}1&2\\5&4\end{matrix}\right] \ni \lambda_1=-1,\lambda_2=6$ then form a matrix $B=\left[\begin{matrix} A & A \\ A & A \end{matrix} \right]$ then try to find the eigen values. But it's not easy >_<

You do understand, I hope, that showing this for this one matrix does NOT show it is true for any square matrix and so does not answer the question?

Look closely at Opalg's very good suggestion.