AppendElement (v,elt)

Append an element to a vector and return the vector. No expansion is done. Normally a row vector is built if starting from null or a 1-by-1 matrix, but if given a column vector it will properly build a column vector.

ApplyOverMatrix (a,func)

Apply a function over all entries of a matrix and return a matrix of the results.

ApplyOverMatrix2 (a,b,func)

Apply a function over all entries of 2 matrices (or 1 value and 1 matrix) and return a matrix of the results.

ColumnsOf (M)

Gets the columns of a matrix as a horizontal vector.

ComplementSubmatrix (m,r,c)

Remove column(s) and row(s) from a matrix.

CompoundMatrix (k,A)

Calculate the kth compound matrix of A.

CountZeroColumns (M)

Count the number of zero columns in a matrix. For example, once you column-reduce a matrix, you can use this to find the nullity. See cref and Nullity.

DeleteColumn (M,col)

Delete a column of a matrix.

DeleteRow (M,row)

Delete a row of a matrix.

DiagonalOf (M)

Gets the diagonal entries of a matrix as a column vector.

See Wikipedia for more information.

DotProduct (u,v)

Get the dot product of two vectors. The vectors must be of the same size. No conjugates are taken so this is a bilinear form even if working over the complex numbers; This is the bilinear scalar product not the sesquilinear scalar product. See HermitianProduct for the standard sesquilinear inner product.

See Wikipedia or Planetmath for more information.

ExpandMatrix (M)

Expands a matrix just like we do on unquoted matrix input. That is we expand any internal matrices as blocks. This is a way to construct matrices out of smaller ones and this is normally done automatically on input unless the matrix is quoted.

HermitianProduct (u,v)

Aliases: InnerProduct

Get the Hermitian product of two vectors. The vectors must be of the same size. This is a sesquilinear form using the identity matrix.

See Wikipedia or Mathworld for more information.

I (n)

Aliases: eye

Return an identity matrix of a given size, that is n by n. If n is zero, returns null.

See Wikipedia or Planetmath for more information.

IndexComplement (vec,msize)

Return the index complement of a vector of indexes. Everything is one based. For example for vector [2,3] and size 5, we return [1,4,5]. If msize is 0, we always return null.

IsDiagonal (M)

Is a matrix diagonal.

See Wikipedia or Planetmath for more information.

IsIdentity (x)

Check if a matrix is the identity matrix. Automatically returns false if the matrix is not square. Also works on numbers, in which case it is equivalent to x==1. When x is null (we could think of that as a 0 by 0 matrix), no error is generated and false is returned.

IsLowerTriangular (M)

Is a matrix lower triangular. That is, are all the entries above the diagonal zero.

IsMatrixInteger (M)

Check if a matrix is a matrix of integers (non-complex).

IsMatrixNonnegative (M)

Check if a matrix is non-negative, that is if each element is non-negative. Do not confuse positive matrices with positive semi-definite matrices.

See Wikipedia for more information.

IsMatrixPositive (M)

Check if a matrix is positive, that is if each element is positive (and hence real). In particular, no element is 0. Do not confuse positive matrices with positive definite matrices.

See Wikipedia for more information.

IsMatrixRational (M)

Check if a matrix is a matrix of rational (non-complex) numbers.

IsMatrixReal (M)

Check if a matrix is a matrix of real (non-complex) numbers.

IsMatrixSquare (M)

Check if a matrix is square, that is its width is equal to its height.

IsUpperTriangular (M)

Is a matrix upper triangular? That is, a matrix is upper triangular if all the entries below the diagonal are zero.

IsValueOnly (M)

Check if a matrix is a matrix of numbers only. Many internal functions make this check. Values can be any number including complex numbers.

IsVector (v)

Is argument a horizontal or a vertical vector. Genius does not distinguish between a matrix and a vector and a vector is just a 1 by n or n by 1 matrix.

IsZero (x)

Check if a matrix is composed of all zeros. Also works on numbers, in which case it is equivalent to x==0. When x is null (we could think of that as a 0 by 0 matrix), no error is generated and true is returned as the condition is vacuous.

LowerTriangular (M)

Returns a copy of the matrix M with all the entries above the diagonal set to zero.

MakeDiagonal (v,arg...)

Aliases: diag

Make diagonal matrix from a vector. Alternatively you can pass in the values to put on the diagonal as arguments. So MakeDiagonal([1,2,3]) is the same as MakeDiagonal(1,2,3).

See Wikipedia or Planetmath for more information.

MakeVector (A)

Alias: MakeColumnVector

Make column vector out of matrix by putting columns above each other. Returns null when given null. Can be used to ensure a vector is a column vector.

MakeRowVector (A)

Make row vector out of matrix by putting rows one after another. Returns null when given null. Can be used to ensure a vector is a row vector.

MatrixProduct (A)

Calculate the product of all elements in a matrix or vector. That is we multiply all the elements and return a number that is the product of all the elements.

MatrixSum (A)

Calculate the sum of all elements in a matrix or vector. That is we add all the elements and return a number that is the sum of all the elements.

MatrixSumSquares (A)

Calculate the sum of squares of all elements in a matrix or vector.

NonzeroColumns (M)

Returns a row vector of the indices of nonzero columns in the matrix M.

Version 1.0.18 onwards.

NonzeroElements (v)

Returns a row vector of the indices of nonzero elements in the vector v.

Version 1.0.18 onwards.

OuterProduct (u,v)

Get the outer product of two vectors. That is, suppose that u and v are vertical vectors, then the outer product is v * u.'.

ReverseVector (v)

Reverse elements in a vector. Return null if given null

RowSum (m)

Calculate sum of each row in a matrix and return a vertical vector with the result.

RowSumSquares (m)

Calculate sum of squares of each row in a matrix and return a vertical vector with the results.

RowsOf (M)

Gets the rows of a matrix as a vertical vector. Each element of the vector is a horizontal vector that is the corresponding row of M. This function is useful if you wish to loop over the rows of a matrix. For example, as for r in RowsOf(M) do something(r).

SetMatrixSize (M,rows,columns)

Make new matrix of given size from old one. That is, a new matrix will be returned to which the old one is copied. Entries that don't fit are clipped and extra space is filled with zeros. If rows or columns are zero then null is returned.

ShuffleVector (v)

Shuffle elements in a vector. Return null if given null.

Version 1.0.13 onwards.

SortVector (v)

Sort vector elements in an increasing order.

StripZeroColumns (M)

Removes any all-zero columns of M.

StripZeroRows (M)

Removes any all-zero rows of M.

Submatrix (m,r,c)

Return column(s) and row(s) from a matrix. This is just equivalent to m@(r,c). r and c should be vectors of rows and columns (or single numbers if only one row or column is needed).

SwapRows (m,row1,row2)

Swap two rows in a matrix.

UpperTriangular (M)

Returns a copy of the matrix M with all the entries below the diagonal set to zero.

columns (M)

Get the number of columns of a matrix.

elements (M)

Get the total number of elements of a matrix. This is the number of columns times the number of rows.

ones (rows,columns...)

Make an matrix of all ones (or a row vector if only one argument is given). Returns null if either rows or columns are zero.

rows (M)

Get the number of rows of a matrix.

zeros (rows,columns...)

Make a matrix of all zeros (or a row vector if only one argument is given). Returns null if either rows or columns are zero.