Diagonal Matrix

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

For example, the following matrix is diagonal:

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

The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form di,i possibly non-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 In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.

Read more about Diagonal Matrix:  Scalar Matrix, Matrix Operations, Other Properties, Uses, Operator Theory

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