1. What Is a Scalar Matrix?
A scalar matrix is a special type of diagonal matrix in which all the diagonal elements are equal. All off-diagonal elements are zero.
If every diagonal entry is the same number \( k \), then the scalar matrix is written as:
\( S = k I_n \)
where \( I_n \) is the identity matrix of order \( n \).
2. Understanding the Structure of a Scalar Matrix
In a scalar matrix:
- The matrix must be square.
- Only the diagonal entries may be non-zero.
- All diagonal entries must be exactly the same number.
This gives it a clean and uniform appearance.
2.1. Example of a Scalar Matrix
\( S = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix} \)
Here, all diagonal entries are 4, so this is a scalar matrix.
2.2. More Examples
- \( \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix} \) — a scalar matrix of order \( 2 \times 2 \).
- \( \begin{bmatrix} -3 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & -3 \end{bmatrix} \) — all diagonal entries are -3.
- \( \begin{bmatrix} 1 \end{bmatrix} \) — a scalar matrix of order \( 1 \times 1 \).
3. How to Identify a Scalar Matrix
To check whether a matrix is a scalar matrix, verify the following:
- Is it a square matrix?
- Are all off-diagonal entries zero?
- Are all diagonal entries equal?
If all three checks are satisfied, the matrix is a scalar matrix.
4. Relation Between Scalar and Other Matrices
A scalar matrix is closely connected to other important matrix types:
- Every scalar matrix is a diagonal matrix.
- Every scalar matrix is also a symmetric matrix.
- The identity matrix is a special scalar matrix with diagonal entries all equal to 1.
This makes scalar matrices a central concept in matrix theory.