10.9. Linear Algebra

AuxilliaryUnitMatrix (n)

Get the auxilliary unit matrix of size n

BilinearForm (v,A,w)

Evaluate (v,w) with respect to the bilinear form given by the matrix A

BilinearFormFunction (A)

Return a function that evaluates two vectors with respect to the bilinear form given by A

CharacteristicPolynomial (M)

Aliases: CharPoly

Get the characteristic polynomial as a vector

CharacteristicPolynomialFunction (M)

Get the characteristic polynomial as a function

ColumnSpace (M)

Get a basis matrix for the columnspace of a matrix

CommutationMatrix (m, n)

Return the commutation matrix K(m,n) which is the unique m*n by m*n matrix such that K(m,n) * MakeVector(A) = MakeVector(A.') for all m by n matrices A.

CompanionMatrix (p)

Companion matrix of a polynomial (as vector)

ConjugateTranspose (M)

Conjugate transpose of a matrix (adjoint). This is the same as the ' operator.

See Planetmath for more information.

Convolution (a,b)

Aliases: convol

Calculate convolution of two horizontal vectors

ConvolutionVector (a,b)

Calculate convolution of two horizontal vectors

CrossProduct (v,w)

CrossProduct of two vectors in R^3

DeterminantalDivisorsInteger (M)

Get the determinantal divisors of an integer matrix (not its characteristic)

DirectSum (M,N...)

Direct sum of matrices

DirectSumMatrixVector (v)

Direct sum of a vector of matrices

Eigenvalues (M)

Aliases: eig

Get the eigenvalues of a square matrix. Currently only works for matrices of size up to 4 by 4, or for triangular matrices (for which the eigenvalues are on the diagonal).

See Planetmath or Mathworld for more information.

GramSchmidt (v,B...)

Apply the Gram-Schmidt process (to the columns) with respect to inner product given by B. If B is not given then the standard hermitian product is used. B can either be a sesquilinear function of two arguments or it can be a matrix giving a sesquilinear form. The vectors will be made orthonormal with respect to B.

See Planetmath for more information.

HankelMatrix (c,r)

Hankel matrix

HilbertMatrix (n)

Hilbert matrix of order n.

See Planetmath for more information.

Image (T)

Get the image (columnspace) of a linear transform

InfNorm (v)

Get the Inf Norm of a vector, sometimes called the sup norm or the max norm.

InvariantFactorsInteger (M)

Get the invariant factors of a square integer matrix (not its characteristic)

InverseHilbertMatrix (n)

Inverse Hilbert matrix of order n.

See Planetmath for more information.

IsHermitian (M)

Is a matrix hermitian. That is, is it equal to its conjugate transpose.

See Planetmath for more information.

IsInSubspace (v,W)

Test if a vector is in a subspace

IsInvertible (n)

Is a matrix (or number) invertible (Integer matrix is invertible iff it's invertible over the integers)

IsInvertibleField (n)

Is a matrix (or number) invertible over a field

IsNormal (M)

Is M a normal matrix. That is, does M*M' == M'*M.

See Planetmath or Mathworld for more information.

IsPositiveDefinite (M)

Is M a hermitian positive definite matrix. That is if HermitianProduct(M*v,v) is always strictly positive for any vector v. M must be square and hermitian to be positive definite. The check that is performed is that every principal submatrix has a nonnegative determinant. (See HermitianProduct)

Note that some authors (for example Mathworld) do not require that M be hermitian, and then the condition is on the real part of the inner product, but we do not take this view. If you wish to perform this check, just check the hermitian part of the matrix M as follows: IsPositiveDefinite(M+M').

See Planetmath or Mathworld for more information.

IsPositiveSemidefinite (M)

Is M a hermitian positive semidefinite matrix. That is if HermitianProduct(M*v,v) is always nonnegative for any vector v. M must be square and hermitian to be positive semidefinite. The check that is performed is that every principal submatrix has a nonnegative determinant. (See HermitianProduct)

Note that some authors do not require that M be hermitian, and then the condition is on the real part of the inner product, but we do not take this view. If you wish to perform this check, just check the hermitian part of the matrix M as follows: IsPositiveSemidefinite(M+M').

See Planetmath or Mathworld for more information.

IsSkewHermitian (M)

Is a matrix skew-hermitian. That is, is the conjugate transpose equal to negative of the matrix.

See Planetmath for more information.

IsUnitary (M)

Is a matrix unitary? That is, does M'*M and M*M' equal the identity.

See Planetmath or Mathworld for more information.

JordanBlock (n,lambda)

Aliases: J

Get the Jordan block corresponding to the eigenvalue lambda with multiplicity n.

See Planetmath or Mathworld for more information.

Kernel (T)

Get the kernel (nullspace) of a linear transform.

(See NullSpace)

LUDecomposition (A, L, U)

Get the LU decomposition of A and store the result in the L and U which should be references. It returns true if successful. For example suppose that A is a square matrix, then after running:

genius> LUDecomposition(A,&L,&U)

This is the LU decomposition of a matrix aka Crout and/or Cholesky reduction. (ISBN 0-201-11577-8 pp.99-103) The upper triangular matrix features a diagonal of values 1 (one). This is not Doolittle's Method which features the 1's diagonal on the lower matrix.

Not all matrices have LU decompositions, for example [0,1;1,0] does not and this function returns false in this case and sets L and U to null.

See Planetmath or Mathworld for more information.

Minor (M,i,j)

Get the i-j minor of a matrix.

See Planetmath for more information.

NonPivotColumns (M)

Return the columns that are not the pivot columns of a matrix

Norm (v,p...)

Aliases: norm

Get the p Norm (or 2 Norm if no p is supplied) of a vector

NullSpace (T)

Get the nullspace of a matrix. That is the kernel of the linear mapping that the matrix represents. This is returned as a matrix whose column space is the nullspace of T.

See Planetmath for more information.

Nullity (M)

Aliases: nullity

Get the nullity of a matrix.

See Planetmath for more information.

OrthogonalComplement (M)

Get the orthogonal complement of the columnspace

PivotColumns (M)

Return pivot columns of a matrix, that is columns which have a leading 1 in row reduced form. Also returns the row where they occur.

Projection (v,W,B...)

Projection of vector v onto subspace W with respect to inner product given by B. If B is not given then the standard hermitian product is used. B can either be a sesquilinear function of two arguments or it can be a matrix giving a sesquilinear form.

QRDecomposition (A, Q)

Get the QR decomposition of a square matrix A, returns the upper triangular matrix R and sets Q to the orthogonal (unitary) matrix. Q should be a reference or null if you don't want any return. For example:

genius> R = QRDecomposition(A,&Q)

See Planetmath or Mathworld for more information.

RayleighQuotient (A,x)

Return the Rayleigh quotient (also called the Rayleigh-Ritz quotient or ratio) of a matrix and a vector.

See Planetmath for more information.

RayleighQuotientIteration (A,x,epsilon,maxiter,vecref)

Find eigenvalues of A using the Rayleigh quotient iteration method. x is a guess at a eigenvector and could be random. It should have nonzero imaginary part if it will have any chance at finding complex eigenvalues. The code will run at most maxiter iterations and return null if we cannot get within an error of epsilon. vecref should either be null or a reference to a variable where the eigenvector should be stored.

See Planetmath for more information on Rayleigh quotient.

Rank (M)

Aliases: rank

Get the rank of a matrix.

See Planetmath for more information.

RosserMatrix ()

Rosser matrix, a classic symmetric eigenvalue test problem

Rotation2D (angle)

Aliases: RotationMatrix

Rotation around origin in R^2

Rotation3DX (angle)

Rotation around origin in R^3 about the x-axis

Rotation3DY (angle)

Rotation around origin in R^3 about the y-axis

Rotation3DZ (angle)

Rotation around origin in R^3 about the z-axis

RowSpace (M)

Get a basis matrix for the rowspace of a matrix

SesquilinearForm (v,A,w)

Evaluate (v,w) with respect to the sesquilinear form given by the matrix A

SesquilinearFormFunction (A)

Return a function that evaluates two vectors with respect to the sesquilinear form given by A

SmithNormalFormField (A)

Smith Normal Form for fields (will end up with 1's on the diagonal)

SmithNormalFormInteger (M)

Smith Normal Form for square integer matrices (not its characteristic)

SolveLinearSystem (M,V,args...)

Solve linear system Mx=V, return solution V if there is a unique solution, null otherwise. Extra two reference parameters can optionally be used to get the reduced M and V.

ToeplitzMatrix (c, r...)

Return the Toeplitz matrix constructed given the first column c and (optionally) the first row r. If only the column c is given then it is conjugated and the nonconjugated version is used for the first row to give a Hermitian matrix (if the first element is real of course).

See Planetmath for more information.

Trace (m)

Aliases: trace

Calculate the trace of a matrix

Transpose (M)

Transpose of a matrix. This is the same as the .' operator.

See Planetmath for more information.

VandermondeMatrix (v)

Aliases: vander

Return the Vandermonde matrix

VectorAngle (v,w,B...)

The angle of two vectors with respect to inner product given by B. If B is not given then the standard hermitian product is used. B can either be a sesquilinear function of two arguments or it can be a matrix giving a sesquilinear form.

VectorSpaceDirectSum (M,N)

The direct sum of the vector spaces M and N

VectorSubspaceIntersection (M,N)

Intersection of the subspaces given by M and N

VectorSubspaceSum (M,N)

The sum of the vector spaces M and N, that is {w | w=m+n, m in M, n in N}

adj (m)

Aliases: Adjugate

Get the classical adjoint (adjugate) of a matrix

cref (M)

Aliases: CREF ColumnReducedEchelonForm

Compute the Column Reduced Echelon Form

det (M)

Aliases: Determinant

Get the determinant of a matrix.

See Planetmath for more information.

ref (M)

Aliases: REF RowEchelonForm

Get the row echelon form of a matrix.

See Planetmath for more information.

rref (M)

Aliases: RREF ReducedRowEchelonForm

Get the reduced row echelon form of a matrix.

See Planetmath for more information.