Scalar Matrix

A scalar matrix is a special type of square matrix in which all the diagonal elements are equal, and all off-diagonal elements are zero. It is a subset of a diagonal matrix where the diagonal elements are the same scalar value.

Mathematically, a scalar matrix \( S \) of order \( n \) is represented as:

\[ S = k I_n = \begin{bmatrix} k & 0 & 0 & \dots & 0 \\ 0 & k & 0 & \dots & 0 \\ 0 & 0 & k & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & k \end{bmatrix} \]

where:

  • \( k \) is a scalar constant.
  • \( I_n \) is the identity matrix of order \( n \).

Properties of Scalar Matrices

1 Commutative Property with Square Matrices

A scalar matrix commutes with any square matrix \( A \) of the same order:

\[ S A = A S \]

Example:

Let

\[ S = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}, \quad A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]

Multiplying:

\[ S A = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix} \]

\[ A S = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix} \]

Since \( S A = A S \), the property holds.

2 Scalar Multiplication

A scalar matrix can be represented as the product of a scalar \( k \) and the identity matrix:

\[ S = k I_n \]

Example:

For \( k = 4 \) and order \( n = 3 \):

\[ S = 4 I_3 = 4 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]

3 Determinant of a Scalar Matrix

The determinant of a scalar matrix of order \( n \) is given by:

\[ \det(S) = k^n \]

Example:

For

\[ S = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix} \]

We compute:

\[ \det(S) = 5^3 = 125 \]

4 Inverse of a Scalar Matrix

If \( k \neq 0 \), the inverse of a scalar matrix is:

\[ S^{-1} = \frac{1}{k} I_n \]

Example:

If

\[ S = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \]

then

\[ S^{-1} = \frac{1}{2} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0.5 & 0 \\ 0 & 0.5 \end{bmatrix} \]

5 Eigenvalues of a Scalar Matrix

The eigenvalues of a scalar matrix are equal to the scalar \( k \), repeated \( n \) times.

Example:

If

\[ S = \begin{bmatrix} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6 \end{bmatrix} \]

The eigenvalues are \( 6, 6, 6 \).

6 Trace of a Scalar Matrix

The trace of a scalar matrix is given by:

\[ \text{Tr}(S) = n \cdot k \]

Example:

If

\[ S = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} \]

The trace is:

\[ \text{Tr}(S) = 2 \times 3 = 6 \]