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  • Determinants

Siblings10
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  • Particular types of matrices
  • Definition of trace of a matrix
  • Cofactors
  • Determinants and operations on matrices
  • Determinants axiomatically
  • The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.
  • The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.
  • Definition of adjugate/classical adjoint of a matrix
  • A matrix is called ill-conditioned if it is nearly singular
  • The condition number of matrix measures how close it is to being singular

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  • Definition of cofactor/submatrix of a matrix
  • Definition of determinant of a matrix as a cofactor expansion across the first row
  • The determinant of a matrix can be computed as a cofactor expansion across any row.
  • The determinant of a matrix can be computed as a cofactor expansion down any column.
  • The inverse of a matrix can be expressed in terms of its matrix of cofactors.
  • Cramer's rule
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  1. Cofactors

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451–475 of 715 nodes
  • Geometric properties of R^n (or C^n)#23795
  • Hermitian matrices#23596
  • Hermitian matrices have real eigenvalues.#23711
  • Homogeneous linear systems are consistent.#23434
  • If A and B are n-by-n matrices#23729
  • If A is a matrix#23679
  • If a matrix has both a left and a right inverse#23519
  • If a set of vectors contains the 0 vector#23825
  • If a set of vectors in R^n (or C^n) contains more than n elements#23827
  • If a space is the direct sum of invariant subspaces#24053
  • If a square matrix has a one-sided inverse#23520
  • If a vector space has dimension n#23838
  • If B is a basis containing b and the b coordinate of c is nonzero#23990
  • If the product of a vector and a scalar is 0#23962
  • If two finite dimensional subspaces have the same dimension and one is contained in the other#24011
  • If two matrices have equal products with all vectors#23484
  • Inner products#24087
  • Inner products in coordinate spaces#23894
  • Inverse#23514
  • Isomorphic vector spaces have the same dimension.#24004
  • Isomorphism#24000
  • Least squares#23945
  • Linear algebra#23410
  • Linear combinations#23799
  • Linear (in)dependence#23815
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