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.
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- Definition of nonsingular matrix: matrix is invertibleDefinition: An $n \times n$ matrix $A$ is nonsingular (or invertible ) if there exists a matrix $A^{-1}$ such that...
- Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solutionDefinition: An $n \times n$ matrix $A$ is nonsingular if the homogeneous system $Ax = 0$ has only the trivial...
- Definition of singular matrix (not nonsingular)Definition: A square matrix $A$ is singular if it is not nonsingular, i.e., it does not have an inverse....
- Proof of several equivalences for nonsingular matrixTheorem: The following are equivalent for an $n \times n$ matrix $A$: 1. $A$ is nonsingular 2. $Ax = 0$ has only the...
- Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.Theorem: An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = b$ has a unique solution for every $b \in...
- Equivalence theorem for nonsingular matrices: the equation Ax=b has a solution for all b.Theorem: An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = b$ has at least one solution for every $b \in...
- Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.Theorem: An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = 0$ has only the trivial solution $x = 0$....
- Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).Theorem: An $n \times n$ matrix $A$ is nonsingular iff the rows of $A$ span $\mathbb{R}^n$ (or $\mathbb{C}^n$). Since...
- Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).Theorem: An $n \times n$ matrix $A$ is nonsingular iff the columns of $A$ span $\mathbb{R}^n$ (or $\mathbb{C}^n$)....
- Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.Theorem: An $n \times n$ matrix $A$ is nonsingular iff the rows of $A$ are linearly independent. For $n$ rows in...
- Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.Theorem: An $n \times n$ matrix $A$ is nonsingular iff the columns of $A$ are linearly independent. This is...
- Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).Theorem: An $n \times n$ matrix $A$ is nonsingular iff the rows of $A$ form a basis for $\mathbb{R}^n$ (or...
- Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).Theorem: An $n \times n$ matrix $A$ is nonsingular iff the columns of $A$ form a basis for $\mathbb{R}^n$ (or...
- Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\dim(\text{Col}(A)) = n$, i.e., $\text{rank}(A) = n$. A...
- Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.Theorem: An $n \times n$ matrix $A$ is nonsingular iff there is a pivot position in every row. With $n$ pivots in an...
- Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ row-reduces to $I n$. This means $A$ is a product of...
- Equivalence theorem for nonsingular matrices: the matrix A has an inverse.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ has an inverse $A^{-1}$ satisfying $AA^{-1} = A^{-1}A =...
- Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ has a left inverse: there exists $B$ such that $BA = I$....
- Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ has a right inverse: there exists $C$ such that $AC = I$....
- Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A^T$ has an inverse. Since $\det(A^T) = \det(A)$, $A^T$ is...
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is one-to-one (injective). $T$ is injective iff...
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is onto (surjective). $T$ is surjective iff the...
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ has an inverse transformation $T^{-1}$. The...
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is an isomorphism. An isomorphism is a bijective...
- Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\det(A) \neq 0$. This follows from $\det(AA^{-1}) = \det(I) =...
- Equivalence theorem for nonsingular matrices: the matrix A has rank n.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\text{rank}(A) = n$. Full rank means the columns span...
- Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\text{null}(A) = \{0\}$. The null space contains only the...
- Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $\text{nullity}(A) = 0$. By the rank-nullity theorem,...
- Equivalence theorem for nonsingular matrices: the matrix A does not have 0 as an eigenvalue.Theorem: An $n \times n$ matrix $A$ is nonsingular iff 0 is not an eigenvalue of $A$. If 0 were an eigenvalue, there...
- Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ is a change-of-basis matrix. A change-of-basis matrix maps...
- Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.Theorem: An $n \times n$ matrix $A$ is nonsingular iff $A$ represents the identity map with respect to some pair of...
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