# 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 |