Math Doubts

Scalar matrix

A matrix that consists of equal diagonal elements and zeros as non-diagonal entries is called a scalar matrix.

Introduction

In a special case, each entry in the main diagonal (or leading diagonal) can be equal and the remaining non-diagonal elements can be zeros in the matrix.

scalar matrix

The following matrix is an example for this case.

$M$ $\,=\,$ $\begin{bmatrix} c & 0 & 0 & \cdots & 0\\ 0 & c & 0 & \cdots & 0\\ 0 & 0 & c & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & & \cdots & c\\ \end{bmatrix} $

In English language, the word “scalar” is used to express “of a quantity”. Hence, the word “scalar” is added before the word “matrix” for expressing the equality of entries in the matrix.

Therefore, a scalar matrix is a matrix in which the diagonal elements are equal and non-zeros but the non-diagonal elements are zeros.

A scalar matrix is basically a square matrix and also a diagonal matrix due to the equality property of the entries.

$e_{11}$ $\,=\,$ $e_{22}$ $\,=\,$ $e_{33}$ $\,=\,$ $\cdots$ $\,=\,$ $e_{mm}$

When the matrix $M$ is simply written as $\begin{bmatrix} e_{ij}\\ \end{bmatrix}$, there are two conditions for calling a matrix as a scalar matrix.

  1. $e_{ij} \,=\, 0$ for all $i \,\ne\, j$
  2. $e_{ij} \,=\, c$ for all $i \,=\, j$ and $c \,\ne\, 0$

Examples

The below three examples are some best examples for a scalar matrix.

$A$ $\,=\,$ $\begin{bmatrix} 6 & 0 \\ 0 & 6 \\ \end{bmatrix} $

It is a square matrix of the order $2$. In this matrix, the diagonal elements are equal and each diagonal element is $6$, and the remaining elements are zero. Hence, the matrix $A$ is called a scalar matrix.

$B$ $\,=\,$ $\begin{bmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1\\ \end{bmatrix} $

The matrix $B$ is a square matrix of the order $3 \times 3$. In this matrix, the diagonal entries are non-zero elements and each entry is $-1$, and the non-diagonal elements are zero. Therefore, the matrix $B$ is an example matrix for a scalar matrix.

$C$ $\,=\,$ $\begin{bmatrix} 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ \end{bmatrix} $

The matrix $C$ is a scalar matrix of the order $4$ because the elements in the main diagonal (or principal diagonal) are non-zeros and each element in the leading diagonal is $2$ but the remaining elements are zero.

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more