scalar matrix
Definition
Noun: A scalar matrix is a specific type of diagonal matrix in which all the elements on the main diagonal are equal to the same scalar value, and all off-diagonal elements are zero.
Usage
A scalar matrix is a fundamental concept in linear algebra. It is used to represent a scaling transformation, where it multiplies all vectors by the same constant factor without changing their direction.
Examples
- In mathematics, the identity matrix is a special case of a scalar matrix where the diagonal elements are all 1.
- The matrix with 3 on the diagonal and zeros elsewhere is a scalar matrix that scales vectors by a factor of 3.
- When a scalar matrix multiplies another matrix, it is equivalent to multiplying that entire matrix by the scalar constant.
Advanced Usage
- Commutative Property: A key property is that a scalar matrix commutes with any other square matrix of the same dimensions (i.e., AB = BA).
- Subspace: The set of all scalar matrices forms a subspace within the vector space of all matrices.
Variants and Related Words
- Diagonal Matrix: A more general matrix where entries outside the main diagonal are zero. Every scalar matrix is a diagonal matrix, but not every diagonal matrix is a scalar matrix.
- Identity Matrix: The multiplicative identity element for matrices, denoted as . It is the scalar matrix with a value of 1 on the diagonal.
- Scalar Multiplication: The operation of multiplying a matrix by a constant (scalar) number, which is analogous to the effect of a scalar matrix.
Synonyms
- Scale matrix
- Homothety matrix (in the context of geometric transformations)
Related Concepts
- Eigenvalue: For a scalar matrix , every vector is an eigenvector with eigenvalue .
- Linear Transformation: A scalar matrix represents a uniform scaling linear transformation.
Noun
-
a diagonal matrix in which all of the diagonal elements are equal
