Scalar Matrix

The scalar matrix is a type of square matrix where the value for all the principal diagonal elements is constant, and the other elements of the matrix are zero. A scalar matrix is formed by multiplying the identity matrix by a constant numerical value. Let us learn more about the definition of the scalar matrix, terms related to the scalar matrix, arithmetic operations of scalar matrix, and the examples of scalar matrix.

What is a Scalar Matrix?

The scalar matrix is a square matrix that has a constant value for every element of its principal diagonal, while all other elements are zero. The scalar matrix has an equal number of rows and columns.

Example:
𝐴 = [
𝑝 0 0
0 𝑝 0
0 0 𝑝]

Scalar Matrix Formula

A scalar matrix is obtained by multiplying an identity matrix by a constant value, i.e.,

Constant × Unity Matrix = Scalar Matrix

In this case, the constant 𝑝 is multiplied by the unity matrix to form the scalar matrix.

𝑥 × [
1 0 0
0 1 0
0 0 1] = [
𝑥 0 0
0 𝑥 0
0 0 𝑥]

Some Important Terms Related to Scalar Matrix

Unity Matrix

The unity matrix is a square matrix and is a multiplicative identity for matrices. The unity matrix contains 1 as its diagonal element and all other elements are equal to zero. Unity matrix has numerous applications in the multiplication of matrices, and in finding the inverse of a matrix. The unity matrix multiplied by a constant value results in a scalar matrix.

𝐼 = [
1 0 0
0 1 0
0 0 1]

Principal Diagonal

The principal diagonal (also called the main diagonal) of a square matrix consists of the elements that extend from the top-left to the bottom-right of the matrix.

Example:
In a 3 × 3 matrix:
[
𝑝 𝑞 𝑟
𝑠 𝑡 𝑢
𝑣 𝑤 𝑥]

The principal diagonal elements are 𝑝, 𝑡, 𝑥.

Diagonal Matrix

The diagonal matrix is a type of square matrix, where each element in the principal diagonal are unique and all other elements are equal to zero. Further, if the diagonal elements of the diagonal matrix are all made equal, then it is called a scalar matrix.

𝐷 = [
𝑝 0 0
0 𝑞 0
0 0 𝑟]

Symmetric Matrix

A symmetric matrix is a type of square matrix, if the transpose of matrix A is equal to the matrix A, i.e., (AT) = A. In other words, let 𝐴 = [𝑎𝑖𝑗]𝑚×𝑛, then A is said to be symmetric, if [𝑎𝑖𝑗] = [𝑎𝑗𝑖], for all possible values of 𝑖 and 𝑗. A scalar matrix is always a symmetric matrix because its transpose is identical to the original matrix.

Example:
Let a matrix A = [
𝑥 0 0
0 𝑥 0
0 0 𝑥]

Transpose of matrix A = (AT) = [
𝑥 0 0
0 𝑥 0
0 0 𝑥] = 𝐴

Zero Matrix

All the elements of a zero matrix are equal to zero.

𝐴 = [
0 0 0
0 0 0
0 0 0]

So, zero matrix is also a scalar matrix because all the elements in the diagonal are equal to 0.

Inverse of Scalar Matrix

The inverse of a scalar matrix is also a scalar matrix whose principal diagonal elements are the reciprocals of the numbers of the original matrix.

Determinant of Scalar Matrix

The determinant of a scalar matrix is equal to the product of the elements on the principal diagonal.

Constant

This is a simple numeric value, which can be an integer, rational number, decimal number, or root value. The identity matrix is multiplied by a constant value to obtain the scalar matrix. A matrix multiplied by a constant value, multiplies with each of the elements of the matrix.

Matrix Operations Using Scalar Matrix

The matrix operations in the scalar matrix are the same as the arithmetic operations in the matrices. But the multiplication of a scalar matrix with another matrix can be observed in the following few steps.

Scalar Matrix 𝑃 = [
𝑘 0 0
0 𝑘 0
0 0 𝑘] , Matrix 𝑄 = [
𝑝 𝑞 𝑟
𝑠 𝑡 𝑢
𝑣 𝑤 𝑥]

𝑃 × 𝑄 = [
𝑘 0 0
0 𝑘 0
0 0 𝑘] × [
𝑝 𝑞 𝑟
𝑠 𝑡 𝑢
𝑣 𝑤 𝑥]

= 𝑘 × [
1 0 0
0 1 0
0 0 1] × [
𝑝 𝑞 𝑟
𝑠 𝑡 𝑢
𝑣 𝑤 𝑥]

= 𝑘 × [
𝑝 𝑞 𝑟
𝑠 𝑡 𝑢
𝑣 𝑤 𝑥]

𝑃 × 𝑄 = [
𝑘𝑝 𝑘𝑞 𝑘𝑟
𝑘𝑠 𝑘𝑡 𝑘𝑢
𝑘𝑣 𝑘𝑤 𝑘𝑥]

𝑃 × 𝑄 = 𝑘𝑄

A scalar matrix is a special type of square matrix where all the elements along the main diagonal are the same scalar value, and all the off-diagonal elements are zero. Scalar matrices play a crucial role in linear algebra, particularly in matrix operations like scalar multiplication, determinants, and eigenvalues. They are a subset of diagonal matrices and are useful in simplifying complex matrix calculations. Understanding scalar matrices helps in grasping fundamental concepts of transformations and linear equations.