📂Matrix Algebra

Toeplitz Matrix

Definition

A component $\left( A \right)_{ij}$ of a matrix $A \in \mathbb{R}^{m \times n}$ is said to satisfy $\left( A \right)_{i, j} = \left( A \right)_{i+1, j+1}$ for all $i, j$ if it is called a Toeplitz matrix. In other words, a Toeplitz matrix is a matrix where all elements along a specific diagonal are the same. $$A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \cdots & a_{-n+1} \\ a_{1} & a_{0} & a_{-1} & \cdots & a_{-n+2} \\ a_{2} & a_{1} & a_{0} & \cdots & a_{-n+3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{m-1} & a_{m-2} & a_{m-3} & \cdots & a_{0} \end{bmatrix}$$

Explanation

A Toeplitz matrix is an extension of a diagonal matrix and frequently appears in numerical analysis or optimization as a tridiagonal matrix.

• Numerical solution of initial value problems for heat equations with Dirichlet boundary conditions
• $\ell_{1}$ Trend Filtering

As seen in the examples above, matrices that incorporate finite differences as such are quite useful. $$D = \begin{bmatrix} 1 & -2 & 1 & & & \\ & 1 & -2 & 1 & & \\ & & \ddots & \ddots & \ddots & \\ & & & 1 & -2 & 1 \end{bmatrix}$$