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Matrices

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Basic terminology and notation
Operations on matrices
Particular types of matrices
Matrix equivalence
Canonical forms of matrices
Factorization of matrices
Similarity of matrices
Nonsingular matrices and equivalences
Eigenvalues and eigenvectors
Determinants
Choosing factorizations and understanding numerical error
Rank and nullity

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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.

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Nonsingular matrices and equivalences

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Equivalence theorem for nonsingular matrices: the matrix A has an inverse.Close match
Definition of nonsingular matrix: matrix is invertibleClose match
Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.Close match
Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.Close match
Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.Close match
Equivalence theorem for nonsingular matrices: the matrix A does not have 0 as an eigenvalue.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.Close match
Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.Close match
Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.Close match
Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.Close match
Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.Close match
Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.Close match
Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.Close match
Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.Close match
Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solutionClose match
Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.Close match

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