Showing two matrices are not unitarily similar

[chuy52506]

Let A=
|1 2|
|0 3|
and B=
|1 0|
|0 3|
Show that A and B are not unitarily similar?

 

[chuy52506]

anyone have an idea of how to do this?

 

[Deveno]

if B = UAU^-1 = UAU* (where U is unitary), then

B* = (UAU*)* = (U*)*A*U* = UA*U*.

but B is symmetric so B = B*. hence we would have

UAU* = UA*U*, and so

U*UAU*U = U*UA*U*U and

A = A*, implying A is symmetric as well, which is false.

 

[g_edgar]

The same thing said in another way: The columns of B are orthogonal, unitary transformation preserves orthogonality, but the columns of A are not orthogonal. Therefore A and B are not unitarily equivalent.

 

[Deveno]

indeed, it all depends on how you define "unitary", by inner product properties, or matrix properties. there is a deep connection between inner products and matrix multiplication.