db48x/doc/commands/matrix.md

Operations with Matrices and vectors

ToArray (→Arry)

Stack to Array Command: Returns a vector of n real or complex elements or a matrix of n × m real or complex elements.

The elements of the result array should be entered in row order.

A1 ... An n ▶ [ A1 ... An ]

A11 ... Arc { r c } ▶ [[ A11 A1c] [ A21 ... Arc ]]

FromArray (Arry→)

Array to Stack Command: Takes an array and returns its elements as separate real or complex numbers. Also returns a list of the dimensions of the array. If the argument is an n-element vector, the first element is returned to level n + 1 (not level nm + 1), and the nth element to level 2.

[ A1 ... An ] ▶ A1 ... An n

[[ A11 A1c] [ A21 ... Arc ]] ▶ A11 ... Arc { r c }

TOCOL

Split an array into column vectors

ADDCOL

Instert a column into an array

REMCOL

Remove a column from an array

FROMCOL

Assemble a matrix from its columns

TODIAG

Extract diagonal elements from a matrix

FROMDIAG

Create a matrix with the given diagonal elements

TOROW

Split an array into its row vectors

ADDROW

Insert a row into an array

REMROW

Remove a row from an array

FROMROW

Assemble an array from its rows

TOV2

Assemble a vector from two values

TOV3

Assemble a vector from three values

FROMV

Split a vector into its elements

AXL

Convert a matrix to list and vice versa

BASIS

Find vectors forming a basis of the subspace represented by the matrix

CHOLESKY

Perform Cholesky decomposition on a matrix

CNRM

Column norm (one norm) of a matrix

CON (ConstantArray)

Returns a constant array, defined as an array whose elements all have the same value.

The constant value is an object taken from argument 2/level 1. The resulting array is either a new array, or an existing array with its elements replaced by the constant, depending on the object in argument 1/level 2.

• Creating a new array: If level 2 contains a list of one or two integers, CON returns a new array. If the list contains a single integer n, CON returns a constant vector with n elements. If the list contains two integers n and m, CON returns a constant matrix with n rows and m columns.

• Replacing the elements of an existing array: If level 2 contains an array, CON returns an array of the same dimensions, with each element equal to the constant.

• If level 2 contains a name, the name must identify a variable that contains a valid input for con, such as an array. In this case, the content of the variable is replaced with the value generated by CON

n k ▶ [ k ... k ]

{ n } k ▶ [ k ... k ]

{ n m } k ▶ [ [ k ... k ] [ k ... k ] ... [ k ... k ] ]

[ vec ] k ▶ [ k ... k]

[ [ mat ] ] k ▶ [ [ k ... k ]]

'name' k ▶

COND

Column norm condition number of a matrix

CROSS

Cross produce of vectors

CSWP

Swap two columns in a matrix

Determinant (DET)

Compute the determinant of a matrix

DIAGMAP

DOT

Internal product (dot product) of vectors

EGV

EGVL

Compute the eigenvalues of a matrix

GRAMSCHMIDT

HADAMARD

Multiply corresponding elements in a matrix

HILBERT

Assemble a Hilbert symbolic array

IBASIS

Find a basis of the intersection of two vector spaces

IDN (IdentityMatrix)

Identity Matrix Command: Returns an identity matrix, that is, a square matrix with its diagonal elements equal to 1 and its off-diagonal elements equal to 0.

The result is either a new square matrix, or an existing square matrix with its elements replaced by the elements of the identity matrix, according to the argument.

• Creating a new matrix: If the argument is an integer n, a new real identity matrix is returned, with its number of rows and number of columns equal to n.

• Replacing the elements of an existing matrix: If the argument is a square matrix, an identity matrix of the same dimensions is returned.

• Generating the identity matrix for a vector: If the argument is a vector with n elements, an identity matrix with n rows and n columns is created.

• If the argument is a name, the name must identify a variable containing on of the valid inputs. In this case, it is replaced with the result.

n ▶ IDN(n)

{ n } ▶ IDN(n)

{ n n } ▶ IDN(n)

[ n-vec ] ▶ IDN(n)

[[ nxn-mat ]] ▶ IDN(n)

'name' ▶

IMAGE

Find a basis of the image of a linear application

ISOM

JORDAN

KER

Find a basis for the kernel of a linear application

LQ

LSQ

LU

LU factorization of a matrix

MAD

MKISOM

PMINI

Minimal polynomial of a matrix

QR

QR Decomposition of a matrix

RANK

Rank of a matrix

RANM (RandomMatrix)

Returns an array containing random integer values between -9 and 9.

RCI

Multiply a row by a constant

RCIJ

Multiply a row by a constant and add to other row

RDM

Change dimensions of an array

REF

Reduce matrix to echelon form (upper triangular form)

RNRM

Row norm (infinity norm) of a matrix

RREF

Fully reduce to row-reduced echelon form

RREFMOD

RSD

Residual R=B-AX' on a system AX=B

RSWP

Swap two rows in a matrix

SCHUR

SNRM

SRAD

SVD

SVL

SYLVESTER

TRACE

Sum of the items in the diagonal of a matrix

TRAN

Transpose a matrix

TRN

Complex conjugate transpose of a matrix

VANDERMONDE

LDUP

Decompose A into LDUP such that PA=LD^-1*U

MMAP

Apply expression or program to the elements of a matrix