# Eigenvalues and Eigenvectors of block tridiagonal Toeplitz matrix.

I have an infinite matrix where all the elements of the diagonal are given by the same $2 \times 2$ real and symmetric matrix , and the elements of the supradiagonal and superdiagonal are the same and given by another $2\times 2$ real and symmetric matrix.

I.e.: $$\begin{bmatrix}A&B\\B&A&B\\&B&A&\ddots\\&&\ddots&\ddots\end{bmatrix},$$ where $A$ and $B$ are $2\times 2$ and real symmetric.

Is it possible to obtain an expression for the eigenvalues and eigenvectors of this matrix?

• Cool question. I'd like to know the answer as well. Commented Jun 21, 2017 at 1:36
• I edited your post to show the matrix using mathjax. I hope you don't mind Commented Jun 21, 2017 at 1:44

Let's call your matrix X, then X is a block Toeplitz symmetric tridiagonal matrix ($$X=T(B,A,B)$$).
Author of https://doi.org/10.1080/09720529.2020.1854939 derived the eigenvalues of $$T(B,A,B)$$ in Theorem 4.1. According to this, the eigenvalues of $$T(B,A,B)$$ are equal to the eigenvalues of $$A+2cos{\frac{j\pi}{N+1}}B$$, in which N is the number of diagonal blocks and $$j=1,2,\cdots,N$$.
Let's define, $$Z(j)=A+2cos{\frac{j\pi}{N+1}}B=\begin{pmatrix} z_1(j) & z_2(j) \\ z_2(j) & z_3(j) \\ \end{pmatrix}$$, which is a $$2\times2$$ matrix. The eigen values of $$Z(j)$$ are $$\frac{z_1(j)+z_3(j)\pm\sqrt{(z_1(j)-z_3(j))^2 +z_2(j)^2}}{2}$$.
Finally, by defining $$A=\begin{pmatrix} a_1 & a_2 \\ a_2 & a_3 \\ \end{pmatrix}$$ and $$B=\begin{pmatrix} b_1 & b_2 \\ b_2 & b_3 \\ \end{pmatrix}$$ we get:
eig(X)=$$\frac{z_1(j)+z_3(j)\pm\sqrt{(z_1(j)-z_3(j))^2 +z_2(j)^2}}{2}$$ for $$j=1,2,\cdots,N$$, where $$z_i(j)=a_i+2cos{\frac{j\pi}{N+1}}b_i$$. In your case, N goes to infinity.
• Thank you for your very helpful answer. Do you know how to find eigenvectors of $T(B,A,B)$? I am also looking at eigenvectors, but the suggested paper does not describe eigenvectors for $T(B,A,B)$. If you have any ideas, I would like to know. Thanks. Commented Jul 17 at 2:33