Question

Let A and B be square matrices of the same size. If A and B are both ______, so is AB.
1) symmetric
2) invertible
4) diagonalisable
8) upper triangular

Answer 1

I know 2 and 8 are correct while 1 is not. Not sure about 4.

Answer 2

For 1) \[(AB)^T = B^T A^T = BA \ne AB \]

Answer 3

that's correct

Answer 4

4 is correct?

Answer 5

hmm ... I have to think about it. what do you think about 2 and 8 ??

Answer 6

most likely no for 4

Answer 7

2 and 8 are correct, I think.
for 2)
(AB)^(-1) (AB)= B^-1 A^-1 (AB) = B^(-1) B = I = AB( B^-1 A^-1 ) = AB ( AB)^(-1)

Answer 8

For 8, we can use block multiplication to prove it

Answer 9

2 is 100% correct

Answer 10

8 is also correct.

Answer 11

about 4 ... do you mean Eiven value decomposition ?

Answer 12

*block multiplication and induction for 8

Answer 13

Hmm, not sure.

Answer 14

could you check your book? most probably it's eiven value decomposition
http://en.wikipedia.org/wiki/Diagonalizable_matrix#How_to_diagonalize_a_matrix

Answer 15

Couldn't find it....
I actually tried this:
\[A = PD_1P^{-1}, B=QD_2Q^{-1}\]Then,\[AB = PD_1P^{-1}QD_2Q^{-1}\]
But this is ugly

Answer 16

it was due to this factor I thought most likely no ... becuase if those two P^-1 and Q are permutation matrices ,, they would wind up the D's and the matrix would not be diagonaliable.

But earlier ... loser66 mentioned that invertible matrix are diagonalizable. and it seems that it's true.

Answer 17

See 7a
http://math.berkeley.edu/~rbayer/s08-54/worksheets/ws414-sol.pdf

Answer 18

A matrix is invertible does not imply that it is diagonalizable

Answer 19

what the hell ... I thought you could eliminate using Gauss Jordan elimination.

Answer 20

let's assume this is true ... and try to find a counter example.

Answer 21

if P^-1Q = I, then it's 100% diagonalizable.

Answer 22

that implies P=Q

Answer 23

Question: The statement is " if both A, B are diagonalizable, then so AB"
It doesn't relate to invertible or not. Am I right?

Answer 24

so, it's true.

Answer 25

Whether A or B is invertible does not guarantee AB is diagonalizable.
Even if a matrix is invertible, it can be not diagonlizable. An counter example can be found in link I posted earlier

Answer 26

I thought pivots are going to act as diagonal elements.