Properties of a Diagonal Matrix

The following are the properties of the Singular Matrix:

• Every diagonal matrix is a square matrix, i.e., a matrix that has an equal number of rows and columns.
• Scalar matrices, identity matrices, and null matrices are examples of diagonal matrices, as their non-principal diagonal elements are zeros.
• The resultant matrix of the sum of two diagonal matrices is also a diagonal matrix.
• The resultant matrix of the product of diagonal matrices is also a diagonal matrix, where the main diagonal elements of the resultant matrix are products of the corresponding elements of the original matrices.

If [Tex]A = \left[\begin{array}{cc} -3 & 0\\ 0 & -3 \end{array}\right]     [/Tex] and [Tex]B = \left[\begin{array}{cc} 8 & 0\\ 0 & 5 \end{array}\right]     [/Tex], then [Tex]AB = \left[\begin{array}{cc} -24 & 0\\ 0 & -15 \end{array}\right]     [/Tex].

• A diagonal matrix is a symmetric matrix, as the transpose of a diagonal matrix is the same matrix itself.

If [Tex]D = \left[\begin{array}{cc} 6 & 0\\ 0 & 7 \end{array}\right]     [/Tex] is a diagonal matrix, then [Tex]D^{T} = \left[\begin{array}{cc} 6 & 0\\ 0 & 7 \end{array}\right]     [/Tex], i.e., D = D^T.

• A diagonal matrix is a symmetric matrix, as the transpose of a diagonal matrix is the same matrix itself.

If [Tex]A = \left[\begin{array}{cc} -5 & 0\\ 0 & 11 \end{array}\right]     [/Tex] and [Tex]B = \left[\begin{array}{cc} 7 & 0\\ 0 & -13 \end{array}\right]     [/Tex] are two diagonal matrices, then

[Tex]A + B = B + A = \left[\begin{array}{cc} 2 & 0\\ 0 & -2 \end{array}\right] [/Tex]

[Tex]AB = BA = \left[\begin{array}{cc} -35 & 0\\ 0 & -143 \end{array}\right] [/Tex]

Learn More,

• Square Matrix
• Null Matrix

Diagonal Matrix is a matrix in which all the non-diagonal elements are zero. A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix.

For example, the order of the matrix that has five rows and four columns is “5 × 4.” We have different types of matrices, such as rectangular, square, triangular, symmetric, singular, etc. The image given below is an “m × n” matrix that has “m” rows and “n” columns.