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Linear Algebra Grinshpan Patterns of matrix multiplication When the number of columns of a matrix A agrees with the number of rows of a matrix B, we may speak of the product matrix AB. The (i, j)-entry of AB is obtained by multiplying together ith row of A and jth column of B : b 1j .. = a b + . . . + a b . (AB)ij = ai1 . . . ain i1 1j in nj . bnj AB has as many rows as A and as many columns as B. The matrix product has many properties and patterns. Some are listed below. These hardly require memorization, but do require some recognition. A row times a column is a number, but a column times a row is a full xa xb x x y a b c = ya yb a b c y = ax + by + cz za zb z z grown matrix: xc yc zc Nonzero matrices may have zero product: 1 −1 1 1 0 0 0 1 0 1 0 0 = = −1 1 1 1 0 0 0 0 0 0 0 0 A zero ∗ ∗ ∗ ∗ 0 0 row of the ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 A zero ∗ ∗ ∗ ∗ ∗ ∗ column ∗ ∗ ∗ ∗ ∗ ∗ first factor is ∗ ∗ ∗ ∗ = ∗ ∗ ∗ 0 0 inherited by the product: ∗ ∗ 0 of the second factor is inherited by the product: 0 ∗ ∗ 0 ∗ 0 ∗ = ∗ 0 ∗ 0 ∗ ∗ 0 ∗ The a 0 0 product of diagonal matrices 0 0 x 0 0 ax 0 b 0 0 y 0 = 0 by 0 c 0 0 z 0 0 is again a diagonal matrix: 0 0 cz The ∗ 0 0 product of upper (lower) triangular matrices is again upper (lower) triangular: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ = 0 ∗ ∗ 0 0 ∗ 0 0 ∗ 0 ∗ Multiplication a 0 0 1 0 b 0 1 0 0 c 1 by a diagonal 1 1 a 1 1 = b 1 1 c matrix on the left has a scaling effect on rows: a a b b c c Multiplication 1 1 1 a 1 1 1 0 1 1 1 0 by a diagonal 0 0 a b 0 = a 0 c a matrix on the right has a scaling effect on columns: b c b c b c Left multiplication by an identity matrix with switched rows switches rows: 0 1 0 a b c d e f 1 0 0 d e f = a b c 0 0 1 g h i g h i Right multiplication by an identity matrix with switched columns switches columns: b a c 0 1 0 a b c d e f 1 0 0 = e d f h g i 0 0 1 g h i Left multiplication by a shear matrix has the corresponding effect on rows: a+d b+e c+f 1 1 0 a b c 0 1 0 d e f = d e f add row 2 to row 1 g h i g h i 0 0 1 a b c 1 0 0 a b c 0 1 0 d e f = d e f subtract row 1 from row 3 −a + g −b + h −c + i g h i −1 0 1 Right multiplication by a shear matrix has the corresponding effect on columns: a b c 1 1 0 a a+b c d e f 0 1 0 = d d + e f add column 1 to column 2 g h i 0 0 1 g g+h i a b c 1 0 0 a + 2c b c d e f 0 1 0 = d + 2f e f add 2 columns 3 to column 1 g h i 2 0 1 g + 2i h i Column-row expansion If C1 , . . . , Cn are the columns of A and R1 , . . . , Rn are the rows of B, then AB = C1 R1 + . . . + Cn Rn . a b a b x y ax + bz ay + bt ax ay bz bt x y + z t = = = + c d c d z t cx + dz cy + dt cx cy dz dt