Question

Couldn't understand what the symbol P^ (P cap ) means in Q5.1 (b) here: http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/transposes-permutations-vector-spaces/MIT18_06SCF11_Ses1.5sol.pdf

Further can't understand the solution: what's the block diagonal matrix? Can't we just use the matrix in 5.1 (a) problem because P^4 = I*P = P not equal to I which is what we need?

Answer 1

P^ is just another variable: you could use Q or R or anything else but Prof Strang likes to use the letter P for permutation matrices. Here he's already used P for a (3,3) matrix so he uses P^ for a (4,4) one.

A block matrix is one where you specify a whole submatrix rather than giving every single element. So [1 0; P 0] here is not a (2,2) matrix but a (4,4) one with the bottom right (3,3) elements equal to P. It turns out you can do matrix multiplication on block matrices by following the usual rules but using matrix multiplication where you have to multiply submatrices. Try it with P^.

Answer 2

Of course I mean [1 0; 0 P]