Understanding Matrix Multiplication Concepts
Explore the four aspects of matrix multiplication, including inner product calculation, combination of columns, and combination of rows, through detailed explanations and visual illustrations.
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Presentation Transcript
Matrix Multiplication Four aspects for multiplication
1. Inner Product (What you have learned in high school)
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 A
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 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
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
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 1 3 1 2 B 5 1 3 5 2 4 6 AB A 17
2. Combination of Columns
Combination of Columns ?? ?? = ? ?1 ?2 ??? = ??1 ??2 ?1 ?2 ?? ?1 ?2 ?? + + + + = ?21 ?11 ?22 ?12 ?2? ?1? ?1 ?1 ?? ?? ?2 ?2 The first column The second column
Combination of Columns 1 3 5 2 4 6 ?? ?? = ? ?1 ?2 1 3 1 2 ??? = ??1 ??2 1 3 5 2 4 6 1 3 5 2 4 6 1 + 3 1 + 2 = The second column The first column
3. Combination of Rows
Combination of Rows ?+ ?12?2 ? + ?1??? ? ?11?1 ? ? ?1 ?2 ?+ ?22?2 ? + ?2??? ?1 ?2 ? ?21?1 ? ? = ? ?? ? ?? ?+ ??2?2 ? + ????? ? ??1?1
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
4. Summation of Matrices
Summation of Matrices ? ?1 ?2 ?3 ? ? ?1 ?2 ?3 ?? ? ?? ?+ ?2?2 ?+ + ???? ? = ?1?1 matrices
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 Block Multiplication
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
