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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?

1 Answers 1

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(homework) so some hints:

  1. The eigenvalues are the roots of ${\rm det}(A-xI) = 0.$

  2. The determinant of a triangular matrix is the product of all diagonal entries.

  3. How many diagonal entries does an $n\times n$ matrix have?

  4. How many roots does $(a - x)^n = 0$ have?

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    Im' glad my answer helped!2012-03-17