6.1. Special Matrix Forms

• For a matrix \(A\), the diagonal elements are the elements \(A_{ij}\) where \(i=j\).

• The off-diagonal elements are elements \(A_{ij}\) such that \(i \neq j\).

• A diagonal is a list of entries \(A_{ij}\) such that \(i - j = k\) is a fixed integer.

• The main diagonal or principal diagonal or primary diagonal or leading diagonal is the list of entries \(A_{ij}\) where \(i = j\).

Consider the matrix:

\[\begin{split}A &= \begin{bmatrix} 1 & 3 & 2\\ 4 & 1 & 3\\ 2 & 5 & 2 \end{bmatrix}\\\end{split}\]

• Its main diagonal is \([1, 1, 2]\).

• The diagonal for \(k=1\) is \([4, 5]\).

• The diagonal for \(k=-1\) is \([3,3]\).

Definition

A diagonal matrix is a matrix whose off-diagonal elements are zero.

Definition

A banded matrix or a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band consisting of the main diagonal and zero or more diagonals on either side.

Definition

A bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above the main diagonal or the diagonal below the main diagonal (but not both). There are exactly two non-zero diagonals.

Definition

An upper bidiagonal matrix is a bidiagonal matrix which has non-zero entries in the diagonal above the main diagonal.

Definition

A lower bidiagonal matrix is a bidiagonal matrix which has non-zero entries in the diagonal below the main diagonal.