Question

how do i show two 2x2 matrices are similar

Answer 1

if the determinant of two matrices are same then they are said to be similar....

Answer 2

if B = X' A X for some nonsingular matrix X, then A and B are said to be similar

Answer 3

hmm, if we can develop a counter example to the det then not :) but i dont recall all the particulars

Answer 4

similar matrixes have the same trace .. but that is not a sufficient condition

Answer 5

oh ok ok i got it...

Answer 6

So if I have two matrices, can I just find the eigenvalues or something to show that they are similar or do I have to actually find X?

Answer 7

The tricky thing is this ... similar matrixes share alot, if not all, the same properties; but .... just because 2 matrixes share some particular properties does not guarantee us they are similar.

Answer 8

B = X A X', or written another way, BX = XA

B = ab X=wx A = mn
cd yz pq


BX = aw + by ax + bz
cw + dy cx + dz

XA = wm + xp wn + xq
ym + zp yn + zq

(a-m)w + by - px + 0z = 0
(a-q)x + bz - wn + 0y = 0
cw + (d-m)y - zp + 0x = 0
cx + (d-q)z - yn + 0w= 0


so, the matrix solution to X=wxyz would be the rref of:

(a-m) -p b 0
-n (a-q) 0 b
c 0 (d-m) p
0 c -n (d-q)

or at least that is what i see it as ... which may mean that it isnt worth a hill of beans :)