Eigenvalues of a special $M \times M$ matrix

Question

I could not obtain an explicit formula for the eigenvalues of matrix $$ \begin{pmatrix} a & b & 0 & 0 & 0 & \cdots & 0 \\ c & a & b & 0 & 0 & \cdots & 0 \\ 0 & c & a & b & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & b \\ 0 & 0 & 0 & \cdots & 0 & c & a \end{pmatrix} $$

It has been claimed that its eigenvalues are $a + 2\sqrt{bc}\cos(\frac{n\pi}{M+1}) \quad n = 1, 2, ..., M$

Any help?

Answer 1

If a matrix has constant diagonals it is called Toeplitz matrix and if a matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal called tridiagonal matrix. So this matrixclass called tridiagonal Toeplitz matrices.

As you have written there is a special form for eigenvalues of tridiagonal Toeplitz matrices.

The eigenvalues and eigenvectors of $$A= \begin{pmatrix} a & b & 0 & 0 & 0 & \cdots & 0 \\ c & a & b & 0 & 0 & \cdots & 0 \\ 0 & c & a & b & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & b \\ 0 & 0 & 0 & \cdots & 0 & c & a \end{pmatrix}, $$ with $a \neq 0 \neq c$ are given by $\lambda_j = b + 2a \sqrt{c/a} \cos \left( \frac{j \pi}{n+1} \right)$ and $$x_j = \begin{pmatrix} (c/a)^{1/2}\sin(1j\pi /(n+1)) \\ (c/a)^{2/2}\sin(2j\pi /(n+1)) \\ (c/a)^{3/2}\sin(3j\pi /(n+1)) \\ \vdots \\ (c/a)^{n/2}\sin(nj\pi /(n+1)) \\ \end{pmatrix} $$ for $j=1,2,\dots,n$, and conclude that $A$ is diagonalizable.

You can find the proof in

C. D. Meyer, Matrix analysis and applied linear algebra, SIAM, 2004.

at pages 514–516. You can find a pdf version the book here.

Answer 2

Hint: For $m=2$ this seems to be correct: Solving $$\left|\begin{pmatrix}a-\lambda&b\\c&a-\lambda \end{pmatrix}\right| =0$$ $$a^2-2a\lambda+\lambda^2-bc=0$$ $$\lambda=a\pm\sqrt{bc}$$ And $\cos\left(\frac{n\pi}{3}\right)=\pm\frac{1}{2}$ for $n=1,2$

Maybe an induction can help...