Nonsingular matrices and equivalences
Nonsingular matrices (also called invertible or nondegenerate matrices) are square matrices that have a multiplicative inverse. They are central to linear algebra because they represent bijective (one-to-one and onto) linear transformations.
An $n \times n$ matrix $A$ is nonsingular if there exists $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$.
The Invertible Matrix Theorem (or Nonsingular Matrix Theorem) provides dozens of equivalent conditions for nonsingularity, connecting concepts from:
- Linear systems ($Ax = b$ has unique solutions)
- Vector spaces (columns/rows form a basis)
- Linear transformations (injectivity, surjectivity, isomorphism)
- Matrix algebra (existence of inverse, nonzero determinant)
- Rank and nullity (full rank, trivial null space)
- Eigenvalues (0 is not an eigenvalue)
The child nodes of this node explore these equivalences in detail.
Parents
Children
- Definition of nonsingular matrix: matrix is invertible
- Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution
- Definition of singular matrix (not nonsingular)
- Proof of several equivalences for nonsingular matrix
- Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.
- Equivalence theorem for nonsingular matrices: the equation Ax=b has a solution for all b.
- Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.
- Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.
- Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.
- Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.
- Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.
- Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.
- Equivalence theorem for nonsingular matrices: the matrix A has an inverse.
- Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.
- Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.
- Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.
- Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.
- Equivalence theorem for nonsingular matrices: the matrix A has rank n.
- Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.
- Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.
- Equivalence theorem for nonsingular matrices: the matrix A does not have 0 as an eigenvalue.
- Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.
- Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.