Inverses and Transpose and permutations
Last updated
Last updated
Example:
For symmetric matrices, A^T = A, so they are unaffected by transposes.
Also R transpose x R is always symetric.
Why? Well let's take the transpose and see if they are the same. (R^T R)^T = R^T x R^TT = R^T x R
Also
Brilliant.org:
Exercise
Permutations P execute row exchanges. P is the identity matrix with reordered rows.
When you want to make row operations, multiply on the left hand side:
When you want to do column operations, multiply on the right:
3x3 Permutations
All the ways I can take the identity matrix and rearrange its rows. For 3x3 matrices there are 6 total permutation matrices. for nxn matrices, there will be n! permutation matrices. Also the inverses are the transposes : P-1 = P T or P T P = I (P transpose x P = Identity matrix).
The first one changes row 1 and 2, the 2nd changes row 1 and 3. The 4th is a cycle, it puts row 2 in row 1 and row 3 in row 1.
The description of elimination WITH row exchanges is PA = LU for any invertable A.
Brilliant.org:
