Let $J = \begin{bmatrix} a & b & 0 & 0 & \cdots & \cdots\\\\ 0 & a & b & 0 & \cdots & \cdots\\\\ \vdots & \vdots & \ddots & \cdots & \cdots & \cdots \\\\ \vdots & \vdots & \vdots & \ddots & \cdots & \cdots \\\\ \vdots & \vdots & \vdots &\ddots & a & b \\\\ \vdots & \vdots & \vdots & \vdots & 0 & a \\ \end{bmatrix}$
I have to find eigenvalues and eigenvectors for $J$.
My thoughts on this...
a = 2 3 0 2 octave-3.2.4.exe:2> b=[2,3,0;0,2,3;0,0,2] b = 2 3 0 0 2 3 0 0 2 octave-3.2.4.exe:3> eig(a) ans = 2 2 octave-3.2.4.exe:4> eig(b) ans = 2 2 2 octave-3.2.4.exe:5>
I can see that the eigenvalue is $a$ for $n \times n$ matrix.
Any idea how I can prove it that is the diagonal for any $N \times N$ matrix.
Thanks!!!
I figured out how to find the eigenvalues. But my eigenvector corresponding to the eigenvalue a comes out to be a zero vector... if I try using matlab, the eigenvector matrix has column vctors with 1 in the first row and zeros in rest of the col vector...
what am I missing? can someone help me figure out that eigenvector matrix?