Transpose.md (1684B)
1 # Transpose 2 3 ML P627 4 5 **Definition:** The transpose of a matrix is the matrix flipped over the diagnol by switching the rows and columns. 6 7 2 4 1 2 3 4 8 3 7 2 -> 4 7 6 9 4 6 3 1 2 3 10 11 As you can see, the first value remains and across the top we have the first column. 12 13 Additionally, the transpose of a vector is possible and will go from n x 1 to 1 x n. 14 15 Example: 16 17 [1 8 0] 18 A = [7 6 4] 19 [2 1 6] 20 21 [1 7 2] 22 A^T = [8 6 1] 23 [0 4 6] 24 25 B = [1 2] 26 [3 4] 27 28 B^T = [1 3] 29 [2 4] 30 31 C = [1 0 -1] 32 [2 7 -5] 33 [4 -3 2] 34 [-1 3 0] 35 36 C^T = [ 1 2 4 -1] 37 [ 0 7 -3 3] 38 [-1 -5 2 0] 39 40 Also, note that (C^T)^T = C. 41 42 #### Determinant 43 44 The determinant of a matrix's transpose is the same as the determinant prior to the transpose. |A| = |A^T| 45 46 Example: 47 48 B = [1 2] 49 [3 4] 50 51 |B| = 4 - 6 = -2 52 53 B^T = [1 3] 54 [2 4] 55 56 |B^T| = 4 - 6 = -2 57 58 #### Product of transpose 59 60 (AB)^T = B^T A^T 61 62 Note: This can be scaled up with an arbitrarily large list of matricies. 63 64 #### Sum of transpose 65 66 Where C = A + B 67 68 C^T = (A+B)^T = A^T + B^T 69 70 This builds upon our knowledge of matrix addition and product of transpose knowledge. 71 72 #### Inverse 73 74 AA^-1 = I_n 75 76 (AA^-1)^T = I_n^T = I_n = (A^-1)^T A^T 77 78 Thus (A^-1)^T is the inverse of A^T. 79 80 Note: the transpose of the identity matrix is still itself as the diagonal does not change. 81 82 #### Vector 83 84 The transpose of a vector is a 1xn matrix where the original vector can be thought of as a nx1 matrix. 85 86 A = [a_1] 87 [a_2] 88 [a_3] 89 90 A^T = [a_1 a_2 a_3] 91 92 #### Column Space 93 94 The transpose of a matrix has a column space equivalent to the row space of the original matrix. 95 96 C(A^T) = Row space(A) 97 98 #### Null Space 99 100 The null space of the transpose is called the left nullspace.