Lesson 4a - permutation matrices
- 2. Learning Intention and Success Criteria Learning Intention: Students will understand that what a Permutation matrix is and how it can be used to rearrange the rows or columns of a matrix Success Criteria: You will be able to create and use Permutation matrices to rearrange the rows and columns of a matrix
- 3. Definition of a Permutation Matrix Permutation Matrix: A matrix, P, such that P is a square matrix made up of only ones and zeros and each row and column have exactly one one. Ex: 𝑃 = 0 0 1 1 0 0 0 1 0 or 𝑃 = 0 1 1 0
- 4. Definition of a Permutation Matrix Notice that I, the identity matrix, is a special case of a Permutation matrix where all of the ones are on the leading diagonal Permutation matrices are similar to the identity matrix in that when you multiply by it, the values don’t change Permutation matrices are different to the identity matrix in that they rearrange the rows or the columns of the matrix.
- 5. How to use Permutation Matrices For some matrix A and permutation matrix 𝑃, 𝑃 × 𝐴 is a row permutation This will rearrange the rows of A Rows always go before columns, so if 𝑃 is before, it's a row permutation 𝐴 × 𝑃 is a column permutation This will rearrange the columns of A
- 6. How to use Permutation Matrices For some matrix A and permutation matrix 𝑃 If 𝑃 is a row permutation matrix, Then, if 𝑝𝑖,𝑗 = 1, row 𝑗 is moved to row 𝑖 If 𝑃 is a column permutation matrix, Then, if 𝑝𝑖,𝑗 = 1, column 𝑖 is moved to column 𝑗
- 7. Using a Permutation Matrix Ex 1: 𝐴 = 1 2 3 4 5 6 , 𝑃 = 0 1 0 0 0 1 1 0 0 , calculate P× 𝐴 and describe the transformations. 𝑃 × 𝐴 = 0 1 0 0 0 1 1 0 0 × 1 2 3 4 5 6 𝑃 × 𝐴 = 3 4 5 6 1 2 Transformations: 𝑝1,2 = 1, so row 2 goes to row 1 𝑝2,3 = 1, so row 3 goes to row 2 𝑝3,1 = 1, so row 1 goes to row 3
- 8. Using a Permutation Matrix Ex 1: 𝐴 = 1 2 3 4 5 6 create a matrix product which results in the matrix 3 1 2 6 4 5 Columns have been rearranged, so we are looking for A × 𝑃 Since A is 2 × 3 , so since P is square P is a (3 × 3) Column 1 goes to column 2, so 𝑝12 = 1 Column 2 goes to column 3, so 𝑝23 = 1 Column 3 goes to column 1, so 𝑝31 = 1 𝐴 × 𝑃 = 1 2 3 4 5 6 0 1 0 0 0 1 1 0 0 𝐴 × 𝑃 = 3 1 2 6 4 5 So P = 0 1 0 0 0 1 1 0 0