# What is the name of matrices of this form?

Consider the upper traingular $N \times N$ matrix

$$\left(\begin{array}{cccccccc} 0 & b_{1} & \dots & b_{q} & 0 & 0 & \dots & 0\\ \vdots & 0 & b_{1} & \dots & b_{q} & 0 & \dots & 0\\ \vdots & & \ddots & & & & & \vdots\\ \vdots & & & 0 & b_{1} & \dots & b_{q} & 0\\ \vdots & & & & 0 & b_{1} & \dots & b_{q}\\ \vdots & & & & & 0 & \dots & 0\\ \vdots & & & & & & \ddots & \vdots\\ 0 & \dots & \dots & \dots & \dots & \dots & \dots & 0 \end{array}\right)$$

Is there a name for matrices of this form?

• "Weird upper triangular matrices..."? – DonAntonio Oct 3 '13 at 17:08

The upper block has several properties that you can combine in a name:

strictly upper triangular, Toeplitz, band matrix with a right/upper bandwidth $q$.

However, if you want to describe the whole matrix, you lose the "Toeplitz" part. You might say

These matrices are of form

$$\begin{bmatrix} X \\ 0 \end{bmatrix}$$

where $0$ is a zero matrix of order $q \times n$, and $X$ is strictly upper triangular, Toeplitz, band matrix of order $(n-q) \times q$ with a right/upper bandwidth $q$.

However, I believe a formal, nameless description is far better.

There is, so far as I know, no standard name for this type of matrix. However, it is composed of the first $q$ superdiagonals of the matrix, so "q-superdiagonal matrix" might be a sensible thing to call them.

Strictly (zeros on the diagonal) upper triangular (obvious) banded (has a limited band) Toeplitz (is a diagonal-constant) matrix.

You can call it a StrUTriBaToM :-)

• That's what I thought at first, but observe the zeroes in the bottom right block. That's why I had to amend my answer. – Vedran Šego Oct 3 '13 at 17:23
• Ah indeed, well then StrUTriBaToM could become StrUTriBATOM for strictly upper triangular banded almost Toeplitz matrix. – Algebraic Pavel Oct 3 '13 at 17:27