Definition

Noun: A real matrix is a rectangular array of numbers arranged in rows and columns, where every single entry (or element) in the array is a real number. Real numbers include all the numbers on the continuous number line, such as integers (e.g., -2, 0, 5), fractions (e.g., 1/2), and irrational numbers (e.g., √2, π).

Usage

The term is used primarily in the fields of linear algebra, mathematics, and computational sciences to specify the type of numbers contained within a matrix. This distinguishes it from other types of matrices, such as complex matrices (which contain complex numbers) or integer matrices.

Examples

• The following is a 2x3 real matrix: [ \begin{bmatrix} 1 & 5.7 & -3 \ 0 & \sqrt{2} & 4.1 \end{bmatrix} ]
• In solving systems of linear equations, the coefficients are often represented in a real matrix.
• The identity matrix, which has 1's on the main diagonal and 0's elsewhere, is a fundamental example of a real matrix.

Advanced Usage

• The set of all real matrices with rows and columns is often denoted as ( M_{m \times n}(\mathbb{R}) ) or ( \mathbb{R}^{m \times n} ), where ( \mathbb{R} ) symbolizes the set of real numbers.
• Operations such as addition, subtraction, and multiplication are defined for real matrices of compatible dimensions. The properties of these operations rely on the underlying properties of real numbers.

Variants and Related Words

• Matrix (noun): The general term for a rectangular array of numbers, symbols, or expressions.
• Complex Matrix (noun): A matrix whose elements are complex numbers.
• Square Matrix (noun): A matrix with the same number of rows and columns. A real matrix can be square or rectangular.
• Real Number (noun): Any rational or irrational number, representing a point on the number line. This is the type of element found in a real matrix.

Synonyms

• Matrix over the reals
• Real-valued matrix

Related Concepts (Not Phrasal Verbs/Idioms)

• Determinant: A scalar value that can be computed from the elements of a square real matrix.
• Eigenvalues and Eigenvectors: Special scalars and vectors associated with a square real matrix.
• Vector Space: The set of all real matrices of a given size forms a vector space over the field of real numbers.

Noun

1. a matrix whose elements are all real numbers