Scalar matrix
From Linear
This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices
Contents
Definition
Verbal definition
A scalar matrix can be defined in the following equivalent ways:
- It is a diagonal matrix where all the diagonal entries are equal to one another.
- It is a scalar multiple of the identity matrix.
Algebraic definition
Suppose 
is a positive integer. A 
matrix 
is termed a scalar matrix if both the following hold:
- Non-diagonal entries are zero:
- Diagonal entries are equal to one another:
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| diagonal matrix | all off-diagonal entries are zero, but the diagonal entries may differ from one another | The matrix  is diagonal but not scalar. |
| | |
| Toeplitz matrix | each row is obtained as a cyclic shift (by one to the right) of the preceding row | A matrix of the form  |
| | |
| symmetric matrix | equals its matrix tranpose | (via diagonal matrix) | (via diagonal matrix) | Bisymmetric matrix, Diagonal matrix|FULL LIST, MORE INFO |
