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

CompanionMatrix (p)

Companion matrix of a polynomial (as vector)

ConjugateTranspose (M)

Conjugate transpose of a matrix (adjoint)

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 matrix (Currently only for 2x2 and triangular matrices)

GramSchmidt (v,B)

Apply the Gram-Schmidt process (to the columns) with respect to innter product given by #2

GramSchmidtDot (M)

Apply the Gram-Schmidt process (to the columns) using the dot product

HankelMatrix (c,r)

Hankel matrix

HilbertMatrix (n)

Hilbert matrix of order n

Image (T)

Get the image (columnspace) of a linear transform

InfNorm (v)

Get the Inf Norm of a vector

InvariantFactorsInteger (M)

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

InverseHilbertMatrix (n)

Inverse Hilbert matrix of order n

IsHermitian (M)

Is a matrix hermitian

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 a matrix normal

IsSkewHermitian (M)

Is a matrix skew-hermitian

IsUnitary (M)

Is a matrix unitary

JordanBlock (n,lambda)

Aliases: J

Get the jordan block corresponding to lambda and n

Kernel (T)

Get the kernel (nullspace) of a linear transform

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 null. 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 Mathworld for more information.

Minor (M,i,j)

Get the i-j minor of a matrix

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

Nullity (M)

Aliases: nullity

Get the nullity of a matrix

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 rref form, also returns the row where they occur

Projection (v,W,B)

Projection of vector v onto subspace W given a bilinear form B

RaleighQuotient (A,x)

Return the Raleigh Quotient of a matrix and a vector

Rank (M)

Aliases: rank

Get the rank of a matrix

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

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.

Trace (m)

Aliases: trace

Calculate the trace of a matrix

Transpose (M)

Transpose of a matrix

VandermondeMatrix (v)

Aliases: vander

Return the Vandermonde matrix

VectorAngle (v,w,B)

The angle of two vectors, given an inner product

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

ref (M)

Aliases: REF RowEchelonForm

Get the row echelon form of a matrix

rref (M)

Aliases: RREF ReducedRowEchelonForm

Get the reduced row echelon form of a matrix