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  • Nonsingular matrices and equivalences

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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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  1. Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.

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  • The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.#23691
  • The eigenvalues of a power of a matrix are the power the eigenvalues.#23690
  • The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.#23689
  • The eigenvalues of a triangular matrix are the entries on the main diagonal.#23709
  • The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.#23692
  • The eigenvectors of a normal matrix are an orthonormal basis.#23602
  • The geometry of linear systems#23427
  • The Gram-Schmidt process converts a linearly independent set into an orthogonal set.#23936
  • The identity matrix is the identity for matrix multiplication.#23498
  • The image of a linearly dependent set under a linear transformation is linearly dependent.#24065
  • The image of a linearly independent set under an injective linear transformation is linearly independent.#24032
  • The inner product of a vector with itself is the square of its norm/length.#23910
  • The intersection of subspaces is a subspace#23967
  • The inverse image of a subspace under a linear transformation is a subspace.#24057
  • The inverse of a linear transformation is a linear transformation#24028
  • The inverse of a matrix can be expressed in terms of its matrix of cofactors.#23721
  • The inverse of a matrix can be used to solve a linear system.#23532
  • The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.#23527
  • The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.#23565
  • The inverse of an isomorphism is an isomorphism.#24002
  • The inverse of a scalar multiple is the reciprocal times the inverse.#23525
  • The inverse of the inverse of a linear transformation is the original linear transformation#24029
  • The kernel/null space of a linear transformation is a subspace#24059
  • The kernels of powers of a linear transformation form an ascending chain#24062
  • The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.#23950
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