What is "diagonal matrix"

n. (context linear algebra English) A matrix in which only the entry on the main diagonal are non-zero.

n. a square matrix with all off-diagonal elements equal to zero

In linear algebra, a diagonal matrix is a matrix (usually a square matrix) in which the off-diagonal elements (↘) are all zero. The main diagonal entries themselves may or may not be zero. Thus, the matrix with n columns and n rows is diagonal if:

d = 0if i ≠ j ∀i, j ∈ {1, 2, …, n}

For example, the following matrix is diagonal:

$$\begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -2\end{bmatrix}$$

The term diagonal matrix may sometimes refer to a , which is an m-by-n matrix with all the entries not of the form d being zero. For example:

$$\begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -3\\ 0 & 0 & 0\\ \end{bmatrix}$$
or $\begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 4 & 0& 0 & 0\\ 0 & 0 & -3& 0 & 0\end{bmatrix}$

However, in the remainder of this article we will consider only square matrices. Any square diagonal matrix is also a symmetric matrix. Also, if the entries come from the field R or C, then it is a normal matrix as well. Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular. The identity matrix I and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.