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I stumbled upon a definition of an nxn square matrix that is called "Toeplitz", where each of the n diagonals consists of their own repeated entries. I am working on a project that involves the property of repeated entries on diagonals, but my matrix is tridiagonal. Can I call this a "tridiagonal Toeplitz" or would that be incorrect?

$$A=\begin{bmatrix} b & a & & & & &\\ c & b & a & & & &\\ & \ddots & \ddots & \ddots & & &\\ & & c & b & a & &\\ & & & \ddots & \ddots & \ddots &\\ & & & & c & b & a\\ & & & & & c & b\\ \end{bmatrix}$$

where $A_{i,i+1}=a$, $A_{i,i}=b$, $A_{i,i-1}=c$.

(I know I don't need the middle part, I updated a matrix that had distinct elements to make this and it takes time to format these)

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    $\begingroup$ Sure, why not? ${}{}{}{}{}{}$ $\endgroup$
    – user1551
    Mar 13, 2018 at 8:41

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Yes, the matrix $A$ is a tridiagonal, Toeplitz matrix. To expand further, there is an explicit, closed-form expression for the eigenvalues of $A$: $$\lambda_k = a - 2\sqrt{bc}\cos\left(\frac{k\pi}{n+1}\right), \quad k=1,\ldots,n.$$ (See ref #1 or #2.)

To expand a little bit more, if you want to solve the system of equations $A \vec{x} = \vec{f}$ when $A$ is a tridiagonal matrix, there's a specific version of Gaussian elimination (the "Thomas algorithm", ref #3) that you can use.

To give a specific example, a tridiagonal Toeplitz matrix $A$ will arise if you consider a "standard" finite difference discretization of the boundary value problem $$u'' = f, \quad u(0)=u(1)=0.$$ In this case, say we have a uniform grid with points $x_j=jh$ (with $j=0, \ldots, n$ and $h = \frac1n$), and denote $u_j \approx u(x_j)$. Then, assuming we discretize $$u''(x_j) = \frac{u_{j+1} - 2u_j + u_{j-1}}{h^2},$$ the linear system of equations will be $A \vec{x} = \vec{f}$, with $a=c=\frac1h$, $b=-\frac2h$, with $\vec{x},\vec{f}\in\mathbb{R}^{n-1}$, and $(\vec{f})_j=f(x_j)$. One reference (out of many possible references) discussing a similar example is on Prof. Li's webpage: http://www4.ncsu.edu/~zhilin/TEACHING/MA402/chapter3.pdf

#1: https://en.wikipedia.org/w/index.php?title=Tridiagonal_matrix&oldid=822910512, Section 1.4 (Eigenvalues)

#2: S. Noschese, L. Pasquini, and L. Reichel (2013), Tridiagonal Toeplitz matrices: properties and novel applications. Numer. Linear Algebra Appl., 20: 302–326. doi:10.1002/nla.1811. (Link to preprint - Eqn (4) in Section 2)

#3: https://en.wikipedia.org/w/index.php?title=Tridiagonal_matrix_algorithm&oldid=826530292

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  • $\begingroup$ Thank you! This is lovely AND the formula for eigenvalues has been derived in my project! ;) $\endgroup$
    – user391838
    Mar 13, 2018 at 16:16

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