Scalar matrix

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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:

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