unit matrix
Definition
- Noun:
- A scalar matrix in which all of the diagonal elements are unity: In linear algebra, a "unit matrix" is a square matrix where all the elements on the main diagonal are equal to one (1), and all other elements are equal to zero (0). It is the multiplicative identity for matrices.
Usage
- The "unit matrix" is fundamental in matrix algebra, serving the same role as the number 1 does in regular multiplication.
- It is often denoted by the capital letter I (for "identity").
- Multiplying any matrix A by the unit matrix (of compatible dimensions) results in the original matrix A (i.e., AI = IA = A).
Examples
- Noun:
- The 3x3 unit matrix is essential for solving this system of linear equations.
- In the equation MI = M, the symbol I represents the appropriate unit matrix.
- A unit matrix is a specific type of diagonal matrix.
Advanced Usage
- "Identity matrix": This is the more common modern term for "unit matrix." The two terms are synonymous in standard usage.
- The identity matrix of size n is often written as Iₙ.
- The concept is central to defining invertible matrices; a matrix A is invertible if there exists a matrix B such that AB = BA = I, where I is the unit matrix.
Variants and Related Words
- Identity matrix (n): The preferred synonym for "unit matrix."
- Scalar matrix (n): A diagonal matrix where all diagonal entries are equal. A unit matrix is a scalar matrix where that scalar is 1.
- Diagonal matrix (n): A matrix where all entries outside the main diagonal are zero. A unit matrix is a special case.
- Singular matrix (n): A matrix that does have an inverse, which is the opposite property of the unit matrix.
Synonyms
- Identity matrix: The standard synonym.
- Multiplicative identity matrix: A more descriptive synonym emphasizing its function.
Related Concepts (Not Phrasal Verbs or Idioms)
- Matrix multiplication: The operation for which the unit matrix acts as the identity element.
- Inverse matrix: A matrix that, when multiplied by the original matrix, yields the unit matrix.
- Determinant: The determinant of a unit matrix is always 1.
Noun
-
a scalar matrix in which all of the diagonal elements are unity
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