Matrix

Since memory is linear and access to memory is alike accesing cells in 1-dimensional array, proper repr of matrix in memory is not possible. However, even if forever stuck with a linear memory, we can pull out a matrix layout, all eith the appropriate methods that accept matrix-like form of indice; they'll get translated into the plain linear indices and that is the area we'retrying to abstract or at least discover useful rules.

In the simplest case, a matrix has 2 dimensions, x and y (in fact, 1D matrix is the simplest and the special case, but it also enrails reffereingto it by the special name, vector).

Translating between indices of array and matrix.
- matrix2d to array, and array to matrix2d
- matrixNd to array
The 9 cell quadrant in the continuous (limitless) table.

array
vector
matrix
tensor

1. INDEX CONVERSIONS

Translating between indices of array and matrix.

Translating between coordinates of (2-dimensional) matrix, M[col,row], into the index of a linear (1-dimensional) array, Array[n].

x is the total number of columns in a matrix.
A[n] is the notation for array.
M[col,row] or M[c][r] is the notation for 2D matrix:
any cell is identified by column, c, and row, r, ordered pair.

Matrix to array

Matrix[c][r] = Array[c + r * x] = Array[n]

Example: Matrix[2][2] = Array[2 + 2 * 4] = Array[10]

Array to matrix

To convert array's index into matrix' col,row coordinates, perform Euclidian division on the index of array, n and x (total nr of columns). So that the pair (quotient, remainder) is (col, row):

Array[n]: (n / x) = (quot, rem) => Matrix[quot = col][rem = rows]

Example: Array[7]: (7/4) = (1,3) => Matrix[1][3]

2. SECTIONS

The 9 cell quadrant and limitless space.

Models

c | RoW| c c c

00 01 02

h v w

c f s

00 01 02

03 04 05

06 07 08

crd - x : y : z CRD - col:row:dep

x = 0 x=1 x=2 x=3

.-----. .-----. .-----. .-----. /002/| / 102 / / 202 / / 302 / d=2 /001/ | / 101 / / 201 / / 301 / d=1 f-----i | f-----i f-----i f-----i d=0 |000| /| |100| |200| |300| r=0 t-----i/ | t-----j t-----j t-----j | ...⊥..| | .-----. | .-----. | .-----. | /012/| | / 112 / | / 212 / | / 312 / d=2 |/011/ | |/ 111 / |/ 211 / |/ 311 / d=1 |-----| | |-----| |-----| |-----| d=0 |010| /| |110| |210| |310| r=1 i-----i/ | i-----i i-----i i-----i | ,..⊥..i | .-----. | .-----. | .-----. | /022/| | / 122 / | / 222 / | / 322 / d=2 |/021/ | |/ 121 / |/ 221 / |/ 321 / d=1 |-----| | |-----| |-----| |-----| d=0 |020| / |120| |220| |320| r=2 L-----i/ L-----j L-----i L-----J ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕

e.g. M[x][y][z] M[c][r][d] M[1][1][1] = x + y*Y + z*Z

000,001,002 | 100,101,102 | 200,201,202 | 300,301,302

ASCII MODELS

                 y
                 𐌣             z
                 |           ↗       
              3 -|         /          
                 |        /             
              2 -|      /               
                 |     /                
              1 -|   /                  
                 |  /                   
-|-------|-------|--|-------|-------|---→ x
-4      -2     0 |  1       3       5 
                 |
            -1  -|
                 |
            -2  -|
                 |
            -3  -|

MATRIX 3D

c | RoW| c c c

00 01 02

h v w

c f s

00 01 02

03 04 05

06 07 08

MATRIX 3D

ARRAY: 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 9 4 b 2 a 1 8 7 0 5 3 6

CHECK: Ma[0, 0] = A[0] Ma[1, 0] = A[1] Ma[2, 0] = A[2] Ma[3, 0] = A[3]

Ma[0, 1] = A[4] Ma[1, 1] = A[5] Ma[2, 1] = A[6] Ma[3, 1] = A[7]

Ma[0, 2] = A[8] Ma[1, 2] = A[9] Ma[2, 2] = A[10] Ma[3, 2] = A[11]

… ⋯ ⋮ ⋱ ⌜ ⌝ ¤ § · ⊻ √ ∝ ∟ ∫ 𐌣 ☇ → ← ↓ ↑ ↗ ↘ ↙ ↖ ↔ ∞ ≈ ≌ ≠ ≤ ≥ ⥽ ∵ ∴ ± ∓ × ÷ ⊕ ⊗ ≡ ⇔ ⇒ ⊢ ⊨ ╳ ∅ ∪ ∩ ⊆ ⊂ ⊃ ∈ ∉ ∋ ∀ ∃ ∄ ¬ ∨ ∧ ⊥ ⊤ ♯ ＃ №

≤ ≥ ÷ ≈ ∞ ‼ ← ° √ ⁿ ² ∞ ≡ ± † ‡ ※ ¶ ¿ § ₪ ¤ ❖ ℃ ℓ ℡ ƒ ฯ Ω Å ㎏㎐㎒㎓

===============================================================

Translating 2D matrix, into a linear array

2D matrix of size R*C is translated into a linear array with R*C cols.
R number of columns in the matrix 4
C num of rows in the matrix 3
matrix rows and columns both start at 0
i iterates over matrix columns i=0..3, i=0..C--
j iterates over matrix rows j=0..2, j=0..R--
m[c][r] -> a[c]
m[0][0] -> a[0] 1st col, 1st row: m[0][0] = 0
m[0][1] -> a[1] 1st col, 2nd row
m[0][2] -> a[2]
m[0][3] -> a[3]
m[0][y] -> a[0]

c o l s | c o l s | c o l s 0 1 2 3 | 4 5 6 7 | 8 9 10 11 1 row

0 1 2 3 | 0 1 2 3 | 0 1 2 3 3 x row 9 4 b 2 | a 1 8 7 | 0 5 3 6 cells

0 1 2 3

4 5 6 7

8 9 a b

9 4 b 2

a 1 8 7

0 5 3 6

c: col iter c_l (first col index), 0 c_h (last col index), 3 C (num of cols), = (c_h + 1 - c_l 1+3-0 = 4 for c = 0:c_l to 3 :c_h

r: row iter r_f: first row, 0 c_l: last row, 2 C = c_l - c_f (num of cols), 2 - 0 = 2 rows for c = c_f to c_l (for c = 0...3)

1. 1. 0 1 2 3 3+1 - 0 = 4
    1 2 3 4 4+1 - 1 = 4
    -5 -4 -3 -2 -2+1 - -5 = 4 = -1 + 5
    getTotalColumns: Hi+1 - Lo

M[c, r] = A[c + r*Ri] Ma[0, 0] = A[0] Ma[1, 0] = A[1] Ma[2, 0] = A[2] Ma[3, 0] = A[3]

Ma[0, 1] = A[4] Ma[1, 1] = A[5] Ma[2, 1] = A[6] Ma[3, 1] = A[7]

Ma[0, 2] = A[8] Ma[1, 2] = A[9] Ma[2, 2] = A[10] Ma[3, 2] = A[11]

(c , r) matrix (r , c) (c + r x R) array (r x R + c) (0, 2) (0 + 2*4) = 8

Case 2: Infinite space, pacman-like-style

Going off of the screen in certian direction, you reappear at the opposite side of the table, moving in the opposite direction.

D4: 𐌣 → ↓ ←, NESW, LRUD d8: 𐌣 ↗ → ↘ ↓ ↙ ← ↖, N NE E SE S SW W NW

        0 1 2 3 4 5 6
    NW        N        NE

0       ↖ . . 𐌣 . . ↗
1       . . . . . . .
2       . . . . . . .
3    W  ← . . . . . →  E
4       . . . . . . .
5       . . . . . . .
6       ↙ . . ↓ . . ↘

    SW        S        SE

. 0 1 2
0 ↖ 𐌣 ↗
1 ← + →
2 ↙ ↓ ↘


NW  N  NE
W   +   E
SW  S  SE

60°40'32"

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Last updated 3 years ago