Matrices¶

CalcpadCE treats matrices as first-class values: a two-dimensional grid of numbers that flows through the same arithmetic, comparison and function calls as scalars and vectors, with row–column rules that keep formulas readable on the worksheet.

The six worksheets in this group walk through the full matrix toolkit step by step. The journey starts with literal construction with | row separators and the indexing/slicing notation, then covers the creational helpers ($matrix$, $identity$, $diagonal$, random, banded) used to assemble structured matrices. Data functions handle row- and column-wise sorting, ordering, lookup and reductions; math functions provide the linear-algebra primitives — Hadamard product, determinant, inverse, transpose, norms and decomposition helpers — and structural functions expose row/column counts, resizing, joining, reshaping and submatrix extraction. The tour closes with matrix and matrix–vector operators, the element-wise versus algebraic multiplication rules, and the comparison and logical operations that broadcast across whole grids.

Matrix Initialization and Indexing¶

Foundations of matrices in CalcpadCE: literal construction with $|$ row separators, single- and double-index access, slicing, submatrix extraction and assignment by index.

Code:

'Direct:
A = [1|2; 3|4; 5; 6|7; 8]
'Direct with vectors:
a = [1; 2; 4]
A = [a|2*a + 1|3*a + 2]
'From a custom function (with vector arguments):
A(x) = transp([x^0|x|x^2|x^3|x^4])
x = [1; 2; 3; 4]
A = A(x)

"Indexing
'<hr/>
A.(3; 2)'- simple dot operator (constant index)
i = 2', 'j = 3
A.(2*i - 1; j + 1)'- expressions with variables
A.(i; j) = 0' - assign a new value to an element:'A
'Initalize values in a block loop
x = [1; 2; 3; 4]
A = matrix(len(x); 7)
#hide
#for i = 1 : n_rows(A)
    #for j = 1 : n_cols(A)
        A.(i; j) = x.i^(j - 1)
    #loop
#loop
#show
A
'Inline loop
B = matrix(len(x); 7)
$Repeat{$Repeat{B.(i; j) = x.i^(j - 1) @ j = 1 : n_cols(B)} @ i = 1 : n_rows(B)}
B

Rendered Output:

Direct:

A = [1 | 2; 3 | 4; 5; 6 | 7; 8] = 100 230 456 780

Direct with vectors:

⃗a = [1; 2; 4] = [1 2 4]

A = [⃗a | 2 · ⃗a + 1 | 3 · ⃗a + 2] = 124 359 5814

From a custom function (with vector arguments):

A ( x ) = transp ( [x⁰ | x | x² | x³ | x⁴] )

⃗x = [1; 2; 3; 4] = [1 2 3 4]

A = A ( ⃗x ) = 11111 124816 1392781 141664256

Indexing

A_3,2 = 3 - simple dot operator (constant index)

i = 2 , j = 3

A_3,4 = 27 - expressions with variables

A_2,3 = 0 - assign a new value to an element: A = 11111 120816 1392781 141664256

Initalize values in a block loop

⃗x = [1; 2; 3; 4] = [1 2 3 4]

A = matrix ( len ( ⃗x ) ; 7 ) = 0000000 0000000 0000000 0000000

A = 1111111 1248163264 1392781243729 14166425610244096

Inline loop

B = matrix ( len ( ⃗x ) ; 7 ) = 0000000 0000000 0000000 0000000

$Repeat{$Repeat{B_i,j = ⃗x_i^{j − 1} for j = 1...n_cols ( B ) } for i = 1...n_rows ( B ) } = 4096

B = 1111111 1248163264 1392781243729 14166425610244096

Matrix Creational Functions¶

Creational helpers for matrices: $matrix$, $identity$, $diagonal$, $utriang$, $ltriang$, $symmetric$, banded, random and Vandermonde constructors.

Code:

'<style>em {font-family:"Times New Roman";}</style>
'<p><b>Matrix</b>(<em>m</em>; <em>n</em>)</p><hr/>
matrix(3; 4)'- creates a 3x4 full (rectangular) matrix

'<p><b>Identity</b>(<em>n</em>)</p><hr/>
identity(3)'- creates  a 3x3 identity matrix (diagonal filled with ones)

'<p><b>Diagonal</b>(<em>n</em>, <em>value</em>)</p><hr/>
diagonal(3; 2)'- creates a 3x3 diagonal matrix filled with 2

'<p><b>Column</b>(<em>m</em>, <em>value</em>)</p><hr/>
column(3; 2)'- creates a 3x1 column matrix filled with 2

'<p><b>Utriang</b>(<em>n</em>, <em>value</em>)</p><hr/>
U = utriang(3)'- creates a 3x3 upper triangular matrix
mfill(U; 1)'- fills the matrix with value of 1

'<p><b>Ltriang</b>(<em>n</em>, <em>value</em>)</p><hr/>
L = ltriang(3)'- creates a 3x3 lower triangular matrix
mfill(L; 1)'- fills the matrix with value of 1

'<p><b>Symmetric</b>(<em>n</em>)</p><hr/>
A = symmetric(4)'- creates a 4x4 empty symmetrix matrix
A.(1; 1) = 5', 'A.(1; 2) = 4'- assign some values
A.(2; 2) = 3', 'A.(2; 3) = 2
A.(4; 2) = 1', 'A.(4; 4) = 1
A'- symmetric values has also changed to preserve symmetry

'<p><b>Vec2diag</b>(<em>vector</em>)</p><hr/>
vec2diag([1; 2; 3])'- creates a 3x3 diagonal matrix from a vector

'<p><b>Vec2col</b>(<em>vector</em>)</p><hr/>
vec2col([1; 2; 3])'- creates a 3x3 column matrix from a vector

'<p><b>Join_Cols</b>(<em>list of vectors</em>)</p><hr/>
join_cols([1]; [4; 5; 6]; [7; 8]; [10; 11; 12])'- creates a general rectangular matrix by joining the input vectors into columns

'<p><b>Join_Rows</b>(<em>list of vectors</em>)</p><hr/>
join_rows([1; 2; 3; 4]; [6; 7; 8; 9; 10])'- creates a general rectangular matrix by joining the input vectors into rows

'<p><b>Augment</b>(<em>list of matrices</em>)</p><hr/>
A = [1; 2|3; 4]
B = [1; 2; 3|4; 5; 6|7; 8; 9]
c = [10; 11; 12; 13]
D = augment(A; B; c)'- creates a general rectangular matrix by joining the input matrices side to side

'<p><b>Stack</b>(<em>list of matrices</em>)</p><hr/>
A = [1; 2|3; 4]
B = [1; 2; 3|4; 5; 6|7; 8; 9]
c = [10; 11]
D = stack(A; B; c)'- creates a general rectangular matrix by joining the input matrices one below the other

Rendered Output:

Matrix(m; n)

matrix ( 3; 4 ) = 0000 0000 0000 - creates a 3x4 full (rectangular) matrix

Identity(n)

identity ( 3 ) = 100 010 001 - creates a 3x3 identity matrix (diagonal filled with ones)

Diagonal(n, value)

diagonal ( 3; 2 ) = 200 020 002 - creates a 3x3 diagonal matrix filled with 2

Column(m, value)

column ( 3; 2 ) = 2 2 2 - creates a 3x1 column matrix filled with 2

Utriang(n, value)

U = utriang ( 3 ) = 000 000 000 - creates a 3x3 upper triangular matrix

mfill ( U; 1 ) = 111 011 001 - fills the matrix with value of 1

Ltriang(n, value)

L = ltriang ( 3 ) = 000 000 000 - creates a 3x3 lower triangular matrix

mfill ( L; 1 ) = 100 110 111 - fills the matrix with value of 1

Symmetric(n)

A = symmetric ( 4 ) = 0000 0000 0000 0000 - creates a 4x4 empty symmetrix matrix

A_1,1 = 5 , A_1,2 = 4 - assign some values

A_2,2 = 3 , A_2,3 = 2

A_4,2 = 1 , A_4,4 = 1

A = 5400 4321 0200 0101 - symmetric values has also changed to preserve symmetry

Vec2diag(vector)

vec2diag ( [1; 2; 3] ) = 100 020 003 - creates a 3x3 diagonal matrix from a vector

Vec2col(vector)

vec2col ( [1; 2; 3] ) = 1 2 3 - creates a 3x3 column matrix from a vector

Join_Cols(list of vectors)

join_cols ( [1]; [4; 5; 6]; [7; 8]; [10; 11; 12] ) = 14710 05811 06012 - creates a general rectangular matrix by joining the input vectors into columns

Join_Rows(list of vectors)

join_rows ( [1; 2; 3; 4]; [6; 7; 8; 9; 10] ) = 12340 678910 - creates a general rectangular matrix by joining the input vectors into rows

Augment(list of matrices)

A = [1; 2 | 3; 4] = 12 34

B = [1; 2; 3 | 4; 5; 6 | 7; 8; 9] = 123 456 789

⃗c = [10; 11; 12; 13] = [10 11 12 13]

D = augment ( A; B; ⃗c ) = 1212310 3445611 0078912 0000013 - creates a general rectangular matrix by joining the input matrices side to side

Stack(list of matrices)

A = [1; 2 | 3; 4] = 12 34

B = [1; 2; 3 | 4; 5; 6 | 7; 8; 9] = 123 456 789

⃗c = [10; 11] = [10 11]

D = stack ( A; B; ⃗c ) = 120 340 123 456 789 1000 1100 - creates a general rectangular matrix by joining the input matrices one below the other

Matrix Data Functions¶

Data-oriented matrix toolkit: row- and column-wise sorting, ordering, search, plus the standard reductions ($sum$, $product$, $min$, $max$, $mean$).

Code:

'<style>em {font-family:"Times New Roman";}</style>
'<p><b>Sort_Cols</b>(<em>matrix</em>; <em>row index</em>) and <b>Rsort_Cols</b>(<em>matrix</em>; <em>row index</em>)</p><hr/>
A = [5; 2; 3|4; 9; 1|6; 8; 7]
B = sort_cols(A; 2)'- the matrix sorted by row 2 in ascending order
C = rsort_cols(A; 2)'- the matrix sorted by row 2 in descending order
A' - the original matrix remains unchanged

'<p><b>Sort_Rows</b>(<em>matrix</em>; <em>column index</em>) and <b>Rsort_Rows</b>(<em>matrix</em>; <em>column index</em>)</p><hr/>
A = [5; 2; 3|4; 9; 1|6; 8; 7]
B = sort_rows(A; 2)'- the matrix sorted by column 2 in ascending order
C = rsort_rows(A; 2)'- the matrix sorted by column 2 in descending order
A' - the original matrix remains unchanged

'<p><b>Order_Cols</b>(<em>matrix</em>; <em>row index</em>) and <b>Revorder_Cols</b>(<em>matrix</em>; <em>row index</em>)</p><hr/>
A = [5; 2; 3|4; 9; 1|6; 8; 7]
b = order_cols(A; 2)'- the indexes of matrix columns ordered by the values in row 2 in ascending order
B = extract_cols(A; b)'- the matrix sorted by row 2 in ascending order
c = revorder_cols(A; 2)'- the indexes of matrix columns ordered by the values in row 2 in descending order
C = extract_cols(A; c)'- the matrix sorted by row 2 in descending order

'<p><b>Order_Rows</b>(<em>matrix</em>; <em>column index</em>) and <b>Revorder_Rows</b>(<em>matrix</em>; <em>column index</em>)</p><hr/>
A = [5; 2; 3|4; 9; 1|6; 8; 7]
b = order_rows(A; 2)'- the indexes of matrix rows ordered by the values in column 2 in ascending order
B = extract_rows(A; b)
c = revorder_rows(A; 2)'- the matrix sorted by row 2 in descending order
C = extract_rows(A; c)'- the indexes of matrix rows ordered by the values in column 2 in descending order

'<p><b>Mcount</b>(<em>matrix</em>; <em>value</em>)</p><hr/>
A = [1; 0; 1|2; 1; 2|1; 3; 1]
n = mcount(A; 1)'- the number of occurances of value 1 in matrix A

'<p><b>Msearch</b>(<em>matrix</em>; <em>value</em>; <em>starting row index</em>; <em>starting column index</em>)</p><hr/>
A = [1; 2; 3|1; 5; 6|1; 8; 9]
b = msearch(A; 1; 1; 1)' - the row and column indexes of the first occurance of value 1 in A, starting from (1,1)
c = msearch(A; 1; 2; 2)' - the same are previous, but starting  from (2,2)
d = msearch(A; 4; 1; 1)' - if the value is not found [0 0] is returned

'<p><b>Mfind</b>(<em>matrix</em>; <em>value</em>)</p><hr/>
A = [1; 2; 3|4; 1; 6|1; 8; 9]
B = mfind(A; 1)' - the row and column indexes of all elements in A, that are equal to 1
C = mfind_ne(A; 1)' - the row and column indexes of all elements in A, that are not equal to 1
D = mfind_lt(A; 5)' - the row and column indexes of all elements in A, that are less than 5
E = mfind(A; 5)'- if the value is not found

'<p><b>Hlookup</b>(<em>matrix</em>; <em>value</em>; <em>reference row</em>; <em>return row</em>)</p><hr/>
A = [0; 1; 0; 1|1; 2; 3; 4; 5|6; 7; 8; 9; 10]
b = hlookup(A; 0; 1; 3)'- the values in row 3, where row 1 contains zeros
c = hlookup_ne(A; 0; 1; 3)'- the values in row 3, where row 1  does not contain zeros
d = hlookup_lt(A; 9; 3; 2)'- the values in row 2, where the respective values in row 3 are &lt; 9
e = hlookup_ge(A; 8; 3; 2)'- the values in row 2, where the respective values in row 3 are &ge; 8
f = hlookup(A; 0; 1; 3)'- the values in row 3, where row 1 contains zeros
g = hlookup(A; 2; 1; 3)'- since 2 is not found in row 1, an empty vector is returned

'<p><b>Vlookup</b>(<em>matrix</em>; <em>value</em>; <em>reference column</em>; <em>return column</em>)</p><hr/>
A = [1; 2|3; 4; 1|5; 6|7; 8; 1|9; 10]
b = vlookup(A; 0; 3; 1)'- the values in column 1, where column 3 contains zeros
c = vlookup_ne(A; 0; 3; 2)'- the values in column 2, where column 3 does not contain zeros
d = vlookup_le(A; 5; 1; 2)'- the values in column 2, where the respective values in column 1 are &le; 5
e = vlookup_gt(A; 3; 1; 2)'- the values in column 2, where the respective values in column 1 are &gt; 3
f = vlookup(A; 2; 3; 1)'- since 2 is not found in column 3, an empty vector is returned

Rendered Output:

Sort_Cols(matrix; row index) and Rsort_Cols(matrix; row index)

A = [5; 2; 3 | 4; 9; 1 | 6; 8; 7] = 523 491 687

B = sort_cols ( A; 2 ) = 352 149 768 - the matrix sorted by row 2 in ascending order

C = rsort_cols ( A; 2 ) = 253 941 867 - the matrix sorted by row 2 in descending order

A = 523 491 687 - the original matrix remains unchanged

Sort_Rows(matrix; column index) and Rsort_Rows(matrix; column index)

A = [5; 2; 3 | 4; 9; 1 | 6; 8; 7] = 523 491 687

B = sort_rows ( A; 2 ) = 523 687 491 - the matrix sorted by column 2 in ascending order

C = rsort_rows ( A; 2 ) = 491 687 523 - the matrix sorted by column 2 in descending order

A = 523 491 687 - the original matrix remains unchanged

Order_Cols(matrix; row index) and Revorder_Cols(matrix; row index)

A = [5; 2; 3 | 4; 9; 1 | 6; 8; 7] = 523 491 687

⃗b = order_cols ( A; 2 ) = [3 1 2] - the indexes of matrix columns ordered by the values in row 2 in ascending order

B = extract_cols ( A; ⃗b ) = 352 149 768 - the matrix sorted by row 2 in ascending order

⃗c = revorder_cols ( A; 2 ) = [2 1 3] - the indexes of matrix columns ordered by the values in row 2 in descending order

C = extract_cols ( A; ⃗c ) = 253 941 867 - the matrix sorted by row 2 in descending order

Order_Rows(matrix; column index) and Revorder_Rows(matrix; column index)

A = [5; 2; 3 | 4; 9; 1 | 6; 8; 7] = 523 491 687

⃗b = order_rows ( A; 2 ) = [1 3 2] - the indexes of matrix rows ordered by the values in column 2 in ascending order

B = extract_rows ( A; ⃗b ) = 523 687 491

⃗c = revorder_rows ( A; 2 ) = [2 3 1] - the matrix sorted by row 2 in descending order

C = extract_rows ( A; ⃗c ) = 491 687 523 - the indexes of matrix rows ordered by the values in column 2 in descending order

Mcount(matrix; value)

A = [1; 0; 1 | 2; 1; 2 | 1; 3; 1] = 101 212 131

n = mcount ( A; 1 ) = 5 - the number of occurances of value 1 in matrix A

Msearch(matrix; value; starting row index; starting column index)

A = [1; 2; 3 | 1; 5; 6 | 1; 8; 9] = 123 156 189

⃗b = msearch ( A; 1; 1; 1 ) = [1 1] - the row and column indexes of the first occurance of value 1 in A, starting from (1,1)

⃗c = msearch ( A; 1; 2; 2 ) = [3 1] - the same are previous, but starting from (2,2)

⃗d = msearch ( A; 4; 1; 1 ) = [0 0] - if the value is not found [0 0] is returned

Mfind(matrix; value)

A = [1; 2; 3 | 4; 1; 6 | 1; 8; 9] = 123 416 189

B = mfind ( A; 1 ) = 123 121 - the row and column indexes of all elements in A, that are equal to 1

C = mfind_ne ( A; 1 ) = 112233 231323 - the row and column indexes of all elements in A, that are not equal to 1

D = mfind_lt ( A; 5 ) = 111223 123121 - the row and column indexes of all elements in A, that are less than 5

E = mfind ( A; 5 ) = 0 0 - if the value is not found

Hlookup(matrix; value; reference row; return row)

A = [0; 1; 0; 1 | 1; 2; 3; 4; 5 | 6; 7; 8; 9; 10] = 01010 12345 678910

⃗b = hlookup ( A; 0; 1; 3 ) = [6 8 10] - the values in row 3, where row 1 contains zeros

⃗c = hlookup_ne ( A; 0; 1; 3 ) = [7 9] - the values in row 3, where row 1 does not contain zeros

⃗d = hlookup_lt ( A; 9; 3; 2 ) = [1 2 3] - the values in row 2, where the respective values in row 3 are < 9

⃗e = hlookup_ge ( A; 8; 3; 2 ) = [3 4 5] - the values in row 2, where the respective values in row 3 are ≥ 8

⃗f = hlookup ( A; 0; 1; 3 ) = [6 8 10] - the values in row 3, where row 1 contains zeros

⃗g = hlookup ( A; 2; 1; 3 ) = [] - since 2 is not found in row 1, an empty vector is returned

Vlookup(matrix; value; reference column; return column)

A = [1; 2 | 3; 4; 1 | 5; 6 | 7; 8; 1 | 9; 10] = 120 341 560 781 9100

⃗b = vlookup ( A; 0; 3; 1 ) = [1 5 9] - the values in column 1, where column 3 contains zeros

⃗c = vlookup_ne ( A; 0; 3; 2 ) = [4 8] - the values in column 2, where column 3 does not contain zeros

⃗d = vlookup_le ( A; 5; 1; 2 ) = [2 4 6] - the values in column 2, where the respective values in column 1 are ≤ 5

⃗e = vlookup_gt ( A; 3; 1; 2 ) = [6 8 10] - the values in column 2, where the respective values in column 1 are > 3

⃗f = vlookup ( A; 2; 3; 1 ) = [] - since 2 is not found in column 3, an empty vector is returned

Matrix Math Functions¶

Linear-algebra primitives: Hadamard product, determinant, inverse, transpose, trace, rank, $L_p$ and Frobenius norms, plus solver and decomposition helpers.

Code:

'<style>em {font-family:"Times New Roman";}</style>
'<p><b>Hprod</b>(<em>first matrix</em>; <em>second matrix</em>)</p><hr/>
A = [1; 2|3; 4|5; 6]'- first matrix
B = [9; 8|7; 6|5; 4]'- second matrix
C = hprod(A; B)'- The Hadamard product of matrices A and B

'<p><b>Fprod</b>(<em>first matrix</em>; <em>second matrix</em>)</p><hr/>
A = [1; 2|3; 4|5; 6]'- first matrix
B = [9; 8|7; 6|5; 4]'- second matrix
C = fprod(A; B)'- the Frobenius product of matrices A and B

'<p><b>Kprod</b>(<em>first matrix</em>; <em>second matrix</em>)</p><hr/>
A = [1; 2|3; 4]'- first matrix
B = [5; 6; 7|8; 9; 10]'- second matrix
C = kprod(A; B)'- the Kronecker product of matrices A and B

'<p><b>Mnorm_1</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
mnorm_1(A)'- the L1 (Manhattan or taxicab) norm of matrix A

'<p><b>Mnorm</b>(<em>matrix</em>) or <b>Mnorm_2</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
mnorm(A)' or
mnorm_2(A)'- the L2 (spectral) norm of matrix A

'<p><b>Mnorm_е</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
mnorm_e(A)'- the Frobenius (a.k.a. Euclidean) norm of matrix A

'<p><b>Mnorm_i</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
mnorm_i(A)'- the L∞ (infinity) norm of matrix A

'<p><b>Cond_1</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
cond_1(A)'- condition number based on the L1 norm of matrix A

'<p><b>Cond</b>(<em>matrix</em>) or <b>Cond_2</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
cond(A)' or
cond_2(A)'- condition number based on the L2 norm of matrix A

'<p><b>Cond_е</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
cond_e(A)'- condition number based on  the Frobenius norm of matrix A

'<p><b>Cond_i</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
cond_i(A)'- condition number based on the L∞ norm of matrix A

'<p><b>Det</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
det(A)'- the determinant of matrix A

'<p><b>Rank</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
rank(A)'- the rank of matrix A

'<p><b>Trace</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
trace(A)'- the trace of matrix A

'<p><b>Transp</b>(<em>matrix</em>)</p><hr/>
A = [1; 2; 3|4; 5; 6|7; 8; 9]
transp(A)'- the transpose of matrix A

'<p><b>Adj</b>(<em>matrix</em>)</p><hr/>
A = [1; 2|3; 4]
adj(A)'- the adjugate of matrix A
D = det(A)'- the determinant of matrix A
A_I = inverse(A)'- the inverse of matrix A
A_I*D'- the inverse multiplied by the determinant shoud give the adjugate

'<p><b>Cofactor</b>(<em>matrix</em>)</p><hr/>
A = [1; 2|3; 4]
C = cofactor(A)'- the cofactor of matrix A
transp(C)'- the transpose of the cofactor shoud give the adjugate'
adj(A)

'<p><b>Eigenvals</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric(3); 1; 1)'- creates a symmetric matrix by copying a general initializer
eigenvals(A)' - the eigenvalues of matrix A

'<p><b>Eigenvecs</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric(3); 1; 1)'- creates a symmetric matrix by copying a general initializer
eigenvecs(A)' - the eigenvectors of matrix A

'<p><b>Eigen</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric(3); 1; 1)' creates symmetric matrix by copying a general initializer
eigen(A)' - the eigenvalues (first column) and eigenvectors (other columns) of matrix A

'<p><b>Cholesky</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric(3); 1; 1)'- creates a symmetric matrix by copying a general initializer
LT = cholesky(A)'- the upper triangular matrix LT by Cholesky decomposition
L = transp(LT)'- the lower triangular matrix L
L*LT'- check: multiplying both matrices should give the original matrix

'<p><b>LU</b>(<em>matrix</em>)</p><hr/>
 A = [4; 12; -16|12; 37; -43|-16; -43; 98]'- creates general matrix
LU = lu(A)'- the combined matrix by LU decomposition
ind'- the index permutation vector
D = not(identity(3))'- a helper matrix to remove the diagonal elements from L

L = hprod(mfill(ltriang(3); 1); LU)^D'- extracts the lower triangular matrix
U = hprod(mfill(utriang(3); 1); LU)'- extracts the upper triangular matrix
extract_rows(L*U; order(ind))'- check: multiplying both matrices and reordering by the permutation vector should give the original matrix

'<p><b>QR</b>(<em>matrix</em>)</p><hr/>
A = [4; 12; -16|12; 37; -43|-16; -43; 98]'- creates a general matrix

QR = qr(A)'- the combined QR matrix
Q = submatrix(QR; 1; 3; 1; 3)'- extracts the Q matrix
R = submatrix(QR; 1; 3; 4; 6)'- extracts the R matrix
Q*R'- check: multiplying both matrices should give the original matrix

'<p><b>SVD</b>(<em>matrix</em>)</p><hr/>
A = [4; 12; -16|12; 37; -43|-16; -43; 98]'- creates a general matrix
SVD = svd(A)'- the combined UΣV matrix, obtained by singular value decomposition
U = submatrix(SVD; 1; 3; 1; 3)'- extracts the U matrix
V = submatrix(SVD; 1; 3; 5; 7)'- extracts the V matrix
σ = col(SVD; 4)'- extracts the singular values
Σ = vec2diag(σ)'- composes the singular value matrix
'Multiplying U,Σ and V may not give the original matrix A, due to the sign ambiguity problem

'<p><b>Inverse</b>(<em>matrix</em>)</p><hr/>
A = [4; 12; -16|12; 37; -43|-16; -43; 98]'- creates a general matrix
B = inverse(A)'- the inverse of A
A*B'- check: multiplying both matrices should give the identity matrix

'<p><b>Lsolve</b>(<em>matrix</em>, <em>vector</em>)</p><hr/>
A = [8; 6; -4|6; 12; -3|-4; -3; 9]'- creates a general matrix
b = [10; 20; 30]'- the right hand side vector
x = lsolve(A; b)'- the solution vector obtained by LU (LDLT) decomposition
A*x'- check: multiplying the matrix by the solution vector should give the right hand side vector

'<p><b>Clsolve</b>(<em>matrix</em>, <em>vector</em>)</p><hr/>
A = copy([8; 6; -4|6; 12; -3|-4; -3; 9]; _
    symmetric(3); 1; 1)'- creates a symmetric matrix
b = [10; 20; 30]'- the right hand side vector
x = clsolve(A; b)'- the solution vector obtained by Cholesky decomposition
A*x'- check: multiplying the matrix by the solution vector should give the right hand side vector

'<p><b>Msolve</b>(<em>matrix</em>, <em>matrix</em>)</p><hr/>
A = [8; 6; -4|6; 12; -3|-4; -3; 9]'- creates a general coefficients matrix
B = join_cols([10; 20; 30]; [40; 50; 60])'- the right hand side matrix
X = msolve(A; B)'- the solution matrix obtained by LU (LDLT) decomposition
A*X'- check: multiplying the coefficients matrix by the solution matrix should give the right hand side matrix

'<p><b>Cmsolve</b>(<em>matrix</em>, <em>matrix</em>)</p><hr/>
A = copy([8; 6; -4|6; 12; -3|-4; -3; 9]; _
    symmetric(3); 1; 1)'- creates a symmetric coefficients  matrix
B = join_cols([10; 20; 30]; [40; 50; 60])'- the right hand side matrix
X = cmsolve(A; B)'- the solution matrix obtained by Cholesky decomposition
A*X'- check: multiplying the coefficients matrix by the solution matrix should give the right hand side matrix

#rad
'Trigonometric:<hr/>
x = [0; 30; 45|60; 90; 180]*(π/180)
sin(x)'- sine
cos(x)'- cosine
tan(x)'- tangent
csc(x)'- cosecant
sec(x)'- secant
cot(x)'- cotangent

'Hyperbolic:<hr/>
x = [-0.5; 0|0.5; 1]
sinh(x)'- hyperbolic sine
cosh(x)'- hyperbolic cosine
tanh(x)'- hyperbolic tangent
csch(x)'- hyperbolic cosecant
sech(x)'- hyperbolic secant
coth(x)'- hyperbolic cotangent

'Inverse trigonometric:<hr/>
x = [-0.5; 0; 0.5|sqrt(2)/2; sqrt(3)/2; 1]
y = [1; sqrt(3)/2; sqrt(2)/2|0.5; 0; -0.5]
asin(x)'- inverse sine
acos(x)'- inverse cosine
atan(x)'- inverse tangent
atan2(x; y)'- the angle whose tangent is the quotient of y and x
acsc(1/x)'- inverse cosecant
asec(1/x)'- inverse secant
acot(x)'- inverse cotangent

'Inverse hyperbolic:<hr/>
x = [1; 2|4; 8]
asinh(x)'- inverse hyperbolic sine
acosh(x)'- inverse hyperbolic cosine
atanh(x/5)'- inverse hyperbolic tangent
acsch(1/x)'- inverse hyperbolic cosecant
asech(1/x)'- inverse hyperbolic secant
acoth(5/x)'- inverse hyperbolic cotangent

'Logarithmic, exponential and roots:<hr/>
log(x)'- decimal logarithm
ln(x)'- natural logarithm
log_2(x)'- binary logarithm
exp(x)'- natural exponent = eˣ
sqr(x)'- square root
cbrt(x)'- cubic root
root(x; 3)'- n-th root

'Rounding:<hr/>
round(x)'- round to the nearest integer
floor(x)'- round to the smaller integer(towards - ∞)
ceiling(x)'- round to the greater integer(towards + ∞)
trunc(x)'- round to the smaller integer(towards zero)

'Integer:<hr/>
x
y = x + 2
mod(x; y)'- the remainder of an integer division
gcd(y)'- the greatest common divisor of several integers
lcm(y)'- the least common multiple of several integers

'Aggregate and interpolation:<hr />
A = [0; 2; 4|6; 8; 12]
b = [5; 3; 1]
min(4; A; 7; b; 10; 11)'- minimum of multiple values
max(4; A; 7; b; 10; 11)'- maximum of multiple values
sum(4; A; 7; b; 10; 11)'- sum of multiple values
sumsq(4; A; 7; b; 10; 11)'- sum of squares
srss(4; A; 7; b; 10; 11)'- square root of sum of squares
average(4; A; 7; b; 10; 11)'- average of multiple values
product(4; A; 7; b; 10; 11)'- product of multiple values
mean(4; A; 7; b; 10; 11)'- geometric mean
take(5; A)'- the 5-th element from the matrix, linearized to list by rows
line(5.5; A)'- linear interpolation along the linearized matrix
spline(5.5; A)'- Hermite spline interpolation along the linearized matrix
take(2; 2; A)'- the element on row 2 and column 2
line(1.5; 2.5; A)'- double linear interpolation
spline(1.5; 2.5; A)'- double Hermite spline interpolation

Rendered Output:

Hprod(first matrix; second matrix)

A = [1; 2 | 3; 4 | 5; 6] = 12 34 56 - first matrix

B = [9; 8 | 7; 6 | 5; 4] = 98 76 54 - second matrix

C = hprod ( A; B ) = 916 2124 2524 - The Hadamard product of matrices A and B

Fprod(first matrix; second matrix)

A = [1; 2 | 3; 4 | 5; 6] = 12 34 56 - first matrix

B = [9; 8 | 7; 6 | 5; 4] = 98 76 54 - second matrix

C = fprod ( A; B ) = 119 - the Frobenius product of matrices A and B

Kprod(first matrix; second matrix)

A = [1; 2 | 3; 4] = 12 34 - first matrix

B = [5; 6; 7 | 8; 9; 10] = 567 8910 - second matrix

C = kprod ( A; B ) = 567101214 8910161820 151821202428 242730323640 - the Kronecker product of matrices A and B

Mnorm_1(matrix)