Question #1d3d1

1 Answer
Feb 8, 2017

Answer:

See below.

Explanation:

For part 1), let's say that #A = I#.

Well then #I + A A^t = 2 I#, which is invertible. So part 1) cannot be false for all #A#.

For part 2), if we say that #A = -I#, then #A A^t = (-I) (-I)^t = (-1)^(t+1) I#.

IOW, repeated eigenvalues, and for a 2x2, we have eigenvalues #lambda_(1,2) = (-1)^(t+1)#.

So they're real and repeated, and of the same sign, BUT not always non-negative.

Those are the obvious counter-examples I could think of, but they're not that good, tbh.

But the question is so open you can let #A# be the null matrix or any rotation matrix or a cyclic matrix or whatever you like.