# 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). 1. Translating between indices of array and matrix. * matrix2d to array, and array to matrix2d * matrixNd to array 2. 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 | l | 00 01 02 | | ------------------- | ---------------------- | | | | | h v w | | | c f s | | | r0 | `00` `01` `02` | | | | | r1 | `03` `04` `05` | | | | | r2 | `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 | l | 00 01 02 | | ------------------- | ---------------------- | | | | | h v w | | | c f s | | | r0 | `00` `01` `02` | | | | | r1 | `03` `04` `05` | | | | | r2 | `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. 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" ```