PermutationMatrix.md (809B)
1 # Permutation Matrix 2 3 **Source:** Strang Lectures 4 5 **Lecture:** 2 6 7 **Definition:** A permutation matrix is a matrix that when multiplied by exchanges rows of the other matrix. 8 9 Permutation matrices are necessary for LU decomposition and Gaussian elimination because sometimes we find there are 0's in the pivot positions. 10 11 Something interesting about permutation matrices is that P^-1 = P^T. 12 13 #### ROWS 14 15 P [a b] = [c d] 16 [c d] [a b] 17 18 P is the permutation matrix and we can state it as: 19 20 [0 1] 21 [1 0] 22 23 This will flip the rows. 24 25 #### COLUMNS 26 27 P [a b] = [b a] 28 [c d] [d c] 29 30 This is not possible. Instead we would need to state it as follows: 31 32 [a b] * P = [b a] 33 [c d] [d c] 34 35 We could then state P as [0 1] 36 [1 0] 37 38 This is because we can only do row manipulations, not columns.