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-A*X' on a system A*X=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 P*A=L*D<sup>-1</sup>*U


## MMAP
Apply expression or program to the elements of a matrix