| Matrix multiplication | AB=C -> C_{ij} = Sigma(n_l=1) a_{il}b_{lj}= +a_{i1}b_{1j}...+a_{in}b_{nj}(i=1...k;j=1...p)when order-A=(k x n) orderB(n x p) |
| Matrix transpose | 1.st row becomes 1.st col, 2.nd row -> 2.nd col ... |
| Matrix transpose properties | |
| (-1) | (A^T)^T=A |
| (-2) | (lambdaA)^T=lambdaA^T |
| (-3) | (A+B)^T=A^T+B^T |
| (-4) | (A+B+C)^T=A^T+B^T+C^T |
| (-5)!! | (AB)^T=B^TA^T |
| (-6)!! | (ABC)^T=C^TB^TA^T |
| Row reduced form when | note: a zero row is a row that contains only zeroes [0000] |
| (R1) | when matrix contains both nonzero- and zeroe-rows, all zeo-rows apear under all nonzero-rows |
| (R2) | the first nonzero element in any nonzero-row is of value 1: [0 0 1 4 3] |
| (R3) | All elements directly below the first nonzero element are 0 , the column has only zeroes below first nonzero element |
| (R4) | the first nonzero element in any nonzero-row is left of previous first nonzero elements: r1[1234] r2[0123] r3[0012] |
| Diagonal matrix | only the diagonal contains nonzero numbers: r1[100]r2[060]r3[002] |
| Zero matrix | all zeroes also the diagonal: r1[000]r2[000]r3[000] |
| Identity matrix I | the diagonal matrix, containing only 1's in its diagonal: r1[100]r2[010]r3[001] |
| (-1) | AI=A |
| (-2) | IA=A |
| Symmetrix matrix | A=A^T : r1[123]r2[245]r3[356] |
| Skew symmetrix matrix | -A=A^T : r1[0 2 -3]r2[-2 0 1]r3[3 -1 0] |
| Lower triangular matrix | a_{ij}=0 for j>i |
| (-1) | product of two Lower triangular matrices is also a Lower triangular matrix |
| Upper triangular matrix | a_{ij}=0 for i>j |
| (-1) | product of two Upper triangular matrices is also a Upper triangular matrix |