What is a matrix with non-zero values exist at main diagonal and the diagonal above/below called

Question

It's a question about the name. As we know matrix with non-zero values only on main diagonal is called 'diagonal matrix', and matrix with upper/lower-triangular part filled with non-zero values is called 'triangular matrix'. My question is, does there exist a specific name for a matrix like: $$ \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 & 0\\ 1 & 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & 1 & \cdots & 0 & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & 1 & 0\\ 0 & 0 & 0 & \cdots & 1 & 1 \end{bmatrix} $$ or $$ \begin{bmatrix} 1 & 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & 1 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & 1 & 1\\ 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix} $$ which have non-zeros values at the main diagonal and the diagonal above/below.

Answer 1

Yes, they are bidiagonal matrices. The first matrix is a lower bidiagonal matrix, the second matrix is an upper bidiagonal matrix. It is not wrong to say that bidiagonal matrices are banded matrices, but it is not precise as the class of banded matrices includes the class of bidiagonal matrices.

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When your focus is on the sparsity (non-zero) pattern, it is common to use the symbol $\times$ (written as \times) to mark a (possible) non-zero entry. A triangular matrix which has all ones on the main diagonal is said to be unit triangular.

Answer 2

Such matrices are called band matrices. https://en.m.wikipedia.org/wiki/Band_matrix

Answer 3

As I know:

bidiagonal (see https://en.wikipedia.org/wiki/Bidiagonal_matrix);

tridiagonal (see https://en.wikipedia.org/wiki/Tridiagonal_matrix);

band matrix (see https://en.wikipedia.org/wiki/Band_matrix).