Question

Let $M_n$ be the $n \times n$ tridiagonal matrix whose diagonal entries are all equal to 0 and whose sub- and super-diagonal entries all equal 1. (a) Find the eigenvalues and eigenvectors of $M_2$ and $M_3$ directly. (b) Prove that the eigenvalues and eigenvectors of $M_n$ are explicitly given by $$ \lambda_k=2 \cos \frac{k \pi}{n+1}, \quad \mathbf{v}_k=\left(\sin \frac{k \pi}{n+1}, \sin \frac{2 k \pi}{n+1}, \ldots \sin \frac{n k \pi}{n+1}\right)^T, \quad k=1, \ldots, n . $$ How do you know that there are no other eigenvalues?

   Let $M_n$ be the $n \times n$ tridiagonal matrix whose diagonal entries are all equal to 0 and whose sub- and super-diagonal entries all equal 1. (a) Find the eigenvalues and eigenvectors of $M_2$ and $M_3$ directly. (b) Prove that the eigenvalues and eigenvectors of $M_n$ are explicitly given by
$$
\lambda_k=2 \cos \frac{k \pi}{n+1}, \quad \mathbf{v}_k=\left(\sin \frac{k \pi}{n+1}, \sin \frac{2 k \pi}{n+1}, \ldots \sin \frac{n k \pi}{n+1}\right)^T, \quad k=1, \ldots, n .
$$
How do you know that there are no other eigenvalues?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 8, Problem 47

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Step 1

### Part (a): Eigenvalues and Eigenvectors of $M_2$ and $M_3$ **For $M_2$:** $$ M_2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$ ** Show more…

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Let $M_n$ be the $n \times n$ tridiagonal matrix whose diagonal entries are all equal to 0 and whose sub- and super-diagonal entries all equal 1. (a) Find the eigenvalues and eigenvectors of $M_2$ and $M_3$ directly. (b) Prove that the eigenvalues and eigenvectors of $M_n$ are explicitly given by $$ \lambda_k=2 \cos \frac{k \pi}{n+1}, \quad \mathbf{v}_k=\left(\sin \frac{k \pi}{n+1}, \sin \frac{2 k \pi}{n+1}, \ldots \sin \frac{n k \pi}{n+1}\right)^T, \quad k=1, \ldots, n . $$ How do you know that there are no other eigenvalues?

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