Scalar matrix

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

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 $n$ is a positive integer. A $n\times n$ matrix $A=(a_{ij})$ is termed a scalar matrix if both the following hold:

• Non-diagonal entries are zero: $a_{ij}=0\ \forall i\neq j\in \{1,2,\dots ,n\}$
• Diagonal entries are equal to one another: $a_{ii}=a_{jj}\ \forall \ i,j\in \{1,2,\dots ,n\}$

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 ${\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}$ is diagonal but not scalar.	|FULL LIST, MORE INFO
Toeplitz matrix	each row is obtained as a cyclic shift (by one to the right) of the preceding row		A matrix of the form ${\begin{pmatrix}a&b\\b&a\\\end{pmatrix}}$	|FULL LIST, MORE INFO
symmetric matrix	equals its matrix tranpose	(via diagonal matrix)	(via diagonal matrix)	Bisymmetric matrix, Diagonal matrix|FULL LIST, MORE INFO