singular matrix
Học thuậtThân thiện
Definition
Noun: A singular matrix is a square matrix whose determinant is equal to zero. This property means the matrix is not invertible; there is no matrix that, when multiplied by it, yields the identity matrix. In linear algebra, a singular matrix represents a transformation that collapses space, reducing its dimension (e.g., mapping a plane to a line).
Usage
A singular matrix is identified mathematically by its zero determinant. It is a key concept in determining whether a system of linear equations has a unique solution. - The system of equations has no unique solution because its coefficient matrix is a singular matrix. - If you calculate the determinant and find it is zero, you have a singular matrix.
Advanced Usage
- In Linear Systems: A singular coefficient matrix in the equation indicates the system either has no solutions or infinitely many solutions, but never a single, unique solution.
- Rank Deficiency: A matrix is singular if and only if it is rank-deficient; its rank is less than the number of its rows or columns.
- Eigenvalue Zero: A singular matrix has at least one eigenvalue equal to zero.
Variants and Related Words
- Nonsingular Matrix (n): The antonym; a square matrix with a non-zero determinant, meaning it is invertible.
- Determinant (n): A scalar value that can be computed from a square matrix and determines if it is singular.
- Invertible Matrix (n): Synonym for a nonsingular matrix.
- Degenerate Matrix (n): Another term for a singular matrix.
Synonyms
- Non-invertible matrix
- Degenerate matrix
- Singular linear transformation (in the context of the map it represents)
Antonyms
- Nonsingular matrix
- Invertible matrix
- Regular matrix
Noun
- a square matrix whose determinant is zero