Understanding Matrix Multiplication in Mathematics
Learn about matrix multiplication in mathematics from this comprehensive guide. Explore the definition, calculations, and various methods of performing matrix multiplication with examples and explanations. Enhance your understanding of linear algebra concepts and applications.
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Matrix Multiplication Hung-yi Lee
Reference Textbook: Chapter 2.1
Matrix Multiplication Given two matrices A and B, the (i, j)-entry of AB is the inner product of row i of A and column j of B B A ??? ? = ??
Matrix Multiplication Given two matrices A and B, the (i, j)-entry of AB is the inner product of row i of A and column j of B 1 3 5 2 4 6 ? = 1 1 2 ? = 3 1 1 + 3 2 1 1 + 2 2 ? = ?? = 1 3 + 3 4 1 3 + 2 4 1 5 + 3 6 1 5 + 2 6
Matrix Multiplication 4 ways Way 1: inner product Given two matrices A and B, the (i, j)-entry of AB is the inner product of row i of A and column j of B B (i,j)-entry A AB
Matrix Multiplication 4 ways Way 1: inner product 1 3 1 2 Given two matrices A and B, the (i, j)-entry of AB is the inner product of row i of A and column j of B B 5 1 3 5 2 4 6 AB A 17
Matrix Multiplication 4 ways Way 2: Linear combination of columns ?1 ?2 ?? ?1 ?2 ?? + + + + = ?21 ?11 ?22 ?12 ?2? ?1? ?1 ?1 ?? ?? ?2 ?2 The first column The second column
Matrix Multiplication 4 ways Way 2: Linear combination of columns 1 3 5 2 4 6 1 3 1 2 1 3 5 2 4 6 1 3 5 2 4 6 1 + 3 1 + 2 = The second column The first column
Matrix Multiplication 4 ways Way 3: Linear combination of rows ?+ ?12?2 ? + ?1??? ? ?11?1 ? ? ?1 ?2 ?+ ?22?2 ? + ?2??? ?1 ?2 ? ?21?1 ? ? = ? ?? ? ?? ?+ ??2?2 ? + ????? ? ??1?1
Matrix Multiplication 4 ways Way 3: Linear combination of rows 1 1 1 + 2 3 2 The first row 1 3 5 2 4 6 1 3 1 2 3 1 1 + 4 3 2 = The second row 5 1 1 + 6 3 2 The third row
Matrix Multiplication 4 ways Way 4: summation of matrices ? ?1 ?2 ?3 ? ? ?1 ?2 ?3 ?? ? ?? ?+ ?2?2 ?+ + ???? ? = ?1?1 matrices
Matrix Multiplication 4 ways Way 4: summation of matrices 1 3 5 1 x 2 2 4 6 1 3 5 2 4 6 1 3 2 x 1 1 2 + = 3 2 1 1 1 x 1 1 3 5 Rank = ? 1 3 5 6 4 8 + 12 18 Rank = ? = 12
Augmentation and Partition Augment: the augment of A and B is [A B] Partition: 3 7 2 4 8 1 1 5 4 2 6 3 3 7 2 4 8 1 1 5 4 2 6 3 ? = ? = 3 7 2 4 8 1 1 5 4 2 6 3 3 7 2 4 8 1 1 5 4 2 6 3 ? = ? =
Block Multiplication 3 0 2 1 1 1 2 0 0 2 1 0 1 3 5 0 4 2 ? = ? = 1 3 6 1 3 1 ?11 ?21 ?12 ?22 ?11 ?21 ?12 ?22 ? = ? = ?11 ?21 ?12 ?22 ?11 ?21 ?12 ?22 Multiply as the small matrices are scalar ?? = ?11?11+ ?12?21 ?21?11+ ?22?21 ?11?12+ ?12?22 ?21?12+ ?22?22 = Don t switch the order
Block Multiplication 3 0 2 1 1 1 2 0 0 2 1 0 1 3 5 0 4 2 ? = ? = 1 3 6 1 3 1 2 x 2 2 x 2 + + 2 X 2 2 X 1 ?? = 2 x 2 + + 1 X 2 1 X 1
Block Multiplication 1 0 6 0 1 0 0 5 0 0 0 0 5 ?2 ? ? 5?2 6 8 9 ? = ? = ? = 8 9 7 7 ?2 ? ? 5?2 ?2 ? ? 5?2 ?2 6? ? = = ?2 25?2 ?2 ? ? 5?2 ?2 6? ? ?2 ? ?3= ??2 = = 25?2 31? 125?2
Matrix Multiplication - Meaning Multiple Input ? = ?? b2 = c1 A A b1 b1 bp = c2 A b2 c2 cp = c1 ?? ?? = ? ?1 ?2 = cp A bp ??? = ??1 ??2
Matrix Multiplication - Meaning Composition Given two function ? and g, the function ? ? . composition g f. is the ? = ? ? ? = ? ? ? g f g f ? = ? ? ? ? Matrix multiplication is the composition of two linear functions.
Matrix Multiplication - Meaning Composition ? ? ? ? ? ? A ? B ? = ?? ? = ?? ? C ? ? ?
Matrix Multiplication - Meaning 1 0 0 The first column of B ? ? ? ? ?1= ?1 A ??1 B ? = ?? ? = ?? 1 0 0 Input standard matrix The first column of C C ?1 ?1= ? ?
Matrix Multiplication - Meaning 0 1 0 The second column of B ? ? ? ? ?2= ?2 A ??2 B ? = ?? ? = ?? 0 1 0 Input standard matrix The second column of C C ?2 ?2= ? ?
? ? A ? B ? = ?? ? = ?? ? ? = ?? C ? The composition of A and B is ???= ?? ? = ??1 ??2 Matrix Multiplication
Example 1 0 0 = 1 0 1 1 0 0 1 0 1 reflection about the x-axis rotation by 180 R2 R2 R2 1 0 0 1 0 0 ? ? ? 1 1 ? = ?? ? = ?? 1 0 0 1 ? ? reflection about the y-axis
Not Communicative ?? ??
Not Communicative ? ? n ? ? ? A ? B B ? ? A ? n ? ? ? If A and B are matrices, then both AB and BA are defined if and only if A and B are square matrices?
Properties Let A and B be k x m matrices, C be an m x n matrix, and P and Q be n x p matrices For any scalar s, s(AC) = (sA)C = A(sC) (A + B)C = AC + BC C(P+Q)=CP+CQ IkA = A = AIm The product of any matrix and a zero matrix is a zero matrix Power of square matrices: A Mn n, Ak= A A A (k times), and by convention, A1= A, A0= In.
Properties ???: n X k ??: k X n Let A be kxm matrices, C be an mxn matrix, ???=? ???? ???? ???? m X k n X m n X m m X k n X k
Special Matrix Diagonal Matrix Symmetric Matrix ??= ? AATand ATA are square and symmetric ????= ?????= ??? ??? ?= ?????= ???
Practical Issue Let A and B be k x m matrices, C be an m x n matrix, and P and Q be n x p matrices A(CP) = (AC)P p n n m m X n X p Multiplication count:
m=1000 k=1 Practical Issue n=1 p=1000 Let A and B be k x m matrices, C be an m x n matrix, and P and Q be n x p matrices A(CP) = (AC)P C A C A P P k X m n X p m X n X p m X n k X m n X p k X m X n m X n 103 106 P A AC CP k X m X p k X n X p k X m m X p n X p k X n 106 103
Practical Issue - GPU 10000 10000 10000 Multiplying two 10000 X 10000 matrices CPU GPU (GTX 980 Ti) More than 20 times faster
