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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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501–525 of 715 nodes
  • Matrix transpose commutes with matrix inverse.#23526
  • Matrix transpose is an involution.#23504
  • Matrix-vector multiplication is a linear transformation.#23856
  • Matrix-vector product is associative#23486
  • Matrix-vector products#23482
  • Multiplication#23487
  • Multiplication by a change of coordinates matrix converts representations for different bases.#23997
  • Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.#23598
  • Multiplication of block/partitioned matrices#23569
  • Multiplicity#23695
  • Multiplying a row by a scalar multiplies the determinant by that scalar.#23739
  • Nilpotent matrices#23580
  • Non-example of a linear transformation#23850
  • Nonsingular matrices and equivalences#23640
  • Normal matrices#23599
  • Norm and length#23907
  • Notation for entry of matrix#23450
  • Notation for the set of m by n matrices#23453
  • Operations on matrices#23460
  • Orthogonality#23914
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
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