## Linear Algebra

AuxiliaryUnitMatrix
`AuxiliaryUnitMatrix (n)`

Get the auxiliary unit matrix of size `n`. This is a square matrix with that is all zero except the superdiagonal being all ones. It is the Jordan block matrix of one zero eigenvalue.

BilinearForm
`BilinearForm (v,A,w)`

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

BilinearFormFunction
`BilinearFormFunction (A)`

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

CharacteristicPolynomial
`CharacteristicPolynomial (M)`

Aliases: `CharPoly`

Get the characteristic polynomial as a vector. That is, return the coefficients of the polynomial starting with the constant term. This is the polynomial defined by `det(M-xI)`. The roots of this polynomial are the eigenvalues of `M`. See also CharacteristicPolynomialFunction.

CharacteristicPolynomialFunction
`CharacteristicPolynomialFunction (M)`

Get the characteristic polynomial as a function. This is the polynomial defined by `det(M-xI)`. The roots of this polynomial are the eigenvalues of `M`. See also CharacteristicPolynomial.

ColumnSpace
`ColumnSpace (M)`

Get a basis matrix for the columnspace of a matrix. That is, return a matrix whose columns are the basis for the column space of `M`. That is the space spanned by the columns of `M`.

CommutationMatrix
`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
`CompanionMatrix (p)`

Companion matrix of a polynomial (as vector).

ConjugateTranspose
`ConjugateTranspose (M)`

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

Convolution
`Convolution (a,b)`

Aliases: `convol`

Calculate convolution of two horizontal vectors.

ConvolutionVector
`ConvolutionVector (a,b)`

Calculate convolution of two horizontal vectors. Return result as a vector and not added together.

CrossProduct
`CrossProduct (v,w)`

CrossProduct of two vectors in R3 as a column vector.

DeterminantalDivisorsInteger
`DeterminantalDivisorsInteger (M)`

Get the determinantal divisors of an integer matrix.

DirectSum
`DirectSum (M,N...)`

Direct sum of matrices.

DirectSumMatrixVector
`DirectSumMatrixVector (v)`

Direct sum of a vector of matrices.

Eigenvalues
`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).

Eigenvectors
`Eigenvectors (M)`
`Eigenvectors (M, &eigenvalues)`
`Eigenvectors (M, &eigenvalues, &multiplicities)`

Get the eigenvectors of a square matrix. Optionally get also the eigenvalues and their algebraic multiplicities. Currently only works for matrices of size up to 2 by 2.

GramSchmidt
`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`.

HankelMatrix
`HankelMatrix (c,r)`

Hankel matrix, a matrix whose skew-diagonals are constant. `c` is the first row and `r` is the last column. It is assumed that both arguments are vectors and the last element of `c` is the same as the first element of `r`.

HilbertMatrix
`HilbertMatrix (n)`

Hilbert matrix of order `n`.

Image
`Image (T)`

Get the image (columnspace) of a linear transform.

InfNorm
`InfNorm (v)`

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

InvariantFactorsInteger
`InvariantFactorsInteger (M)`

Get the invariant factors of a square integer matrix.

InverseHilbertMatrix
`InverseHilbertMatrix (n)`

Inverse Hilbert matrix of order `n`.

IsHermitian
`IsHermitian (M)`

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

IsInSubspace
`IsInSubspace (v,W)`

Test if a vector is in a subspace.

IsInvertible
`IsInvertible (n)`

Is a matrix (or number) invertible (Integer matrix is invertible if and only if it is invertible over the integers).

IsInvertibleField
`IsInvertibleField (n)`

Is a matrix (or number) invertible over a field.

IsNormal
`IsNormal (M)`

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

IsPositiveDefinite
`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 non-negative 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')`.

IsPositiveSemidefinite
`IsPositiveSemidefinite (M)`

Is `M` a Hermitian positive semidefinite matrix. That is if `HermitianProduct(M*v,v)` is always non-negative 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 non-negative 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')`.

IsSkewHermitian
`IsSkewHermitian (M)`

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

IsUnitary
`IsUnitary (M)`

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

JordanBlock
`JordanBlock (n,lambda)`

Aliases: `J`

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

Kernel
`Kernel (T)`

Get the kernel (nullspace) of a linear transform.

(See NullSpace)

KroneckerProduct
`KroneckerProduct (M, N)`

Aliases: `TensorProduct`

Compute the Kronecker product (tensor product in standard basis) of two matrices.

Version 1.0.18 onwards.

LUDecomposition
`LUDecomposition (A, L, U)`

Get the LU decomposition of `A`, that is find a lower triangular matrix and upper triangular matrix whose product is `A`. 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)`
```

You will have the lower matrix stored in a variable called `L` and the upper matrix in a variable called `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`.

Minor
`Minor (M,i,j)`

Get the `i`-`j` minor of a matrix.

NonPivotColumns
`NonPivotColumns (M)`

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

Norm
`Norm (v,p...)`

Aliases: `norm`

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

NullSpace
`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`.

Nullity
`Nullity (M)`

Aliases: `nullity`

Get the nullity of a matrix. That is, return the dimension of the nullspace; the dimension of the kernel of `M`.

OrthogonalComplement
`OrthogonalComplement (M)`

Get the orthogonal complement of the columnspace.

PivotColumns
`PivotColumns (M)`

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

Projection
`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
`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)`
```

You will have the upper triangular matrix stored in a variable called `R` and the orthogonal (unitary) matrix stored in `Q`.

RayleighQuotient
`RayleighQuotient (A,x)`

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

RayleighQuotientIteration
`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.

Rank
`Rank (M)`

Aliases: `rank`

Get the rank of a matrix.

RosserMatrix
`RosserMatrix ()`

Returns the Rosser matrix, which is a classic symmetric eigenvalue test problem.

Rotation2D
`Rotation2D (angle)`

Aliases: `RotationMatrix`

Return the matrix corresponding to rotation around origin in R2.

Rotation3DX
`Rotation3DX (angle)`

Return the matrix corresponding to rotation around origin in R3 about the x-axis.

Rotation3DY
`Rotation3DY (angle)`

Return the matrix corresponding to rotation around origin in R3 about the y-axis.

Rotation3DZ
`Rotation3DZ (angle)`

Return the matrix corresponding to rotation around origin in R3 about the z-axis.

RowSpace
`RowSpace (M)`

Get a basis matrix for the rowspace of a matrix.

SesquilinearForm
`SesquilinearForm (v,A,w)`

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

SesquilinearFormFunction
`SesquilinearFormFunction (A)`

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

SmithNormalFormField
`SmithNormalFormField (A)`

Returns the Smith normal form of a matrix over fields (will end up with 1's on the diagonal).

SmithNormalFormInteger
`SmithNormalFormInteger (M)`

Return the Smith normal form for square integer matrices over integers.

SolveLinearSystem
`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
`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).

Trace
`Trace (M)`

Aliases: `trace`

Calculate the trace of a matrix. That is the sum of the diagonal elements.

Transpose
`Transpose (M)`

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

VandermondeMatrix
`VandermondeMatrix (v)`

Aliases: `vander`

Return the Vandermonde matrix.

VectorAngle
`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
`VectorSpaceDirectSum (M,N)`

The direct sum of the vector spaces M and N.

VectorSubspaceIntersection
`VectorSubspaceIntersection (M,N)`

Intersection of the subspaces given by M and N.

VectorSubspaceSum
`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`

cref
`cref (M)`

Aliases: `CREF` `ColumnReducedEchelonForm`

Compute the Column Reduced Echelon Form.

det
`det (M)`

Aliases: `Determinant`

Get the determinant of a matrix.

ref
`ref (M)`

Aliases: `REF` `RowEchelonForm`

Get the row echelon form of a matrix. That is, apply gaussian elimination but not backaddition to `M`. The pivot rows are divided to make all pivots 1.

`rref (M)`
Aliases: `RREF` `ReducedRowEchelonForm`
Get the reduced row echelon form of a matrix. That is, apply gaussian elimination together with backaddition to `M`.