Showing two matrices are not unitarily similar

  • Thread starter chuy52506
  • Start date
  • #1
77
0

Main Question or Discussion Point

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

Answers and Replies

  • #2
77
0
anyone have an idea of how to do this?
 
  • #3
Deveno
Science Advisor
906
6
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.
 
  • #4
607
0
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.
 
  • #5
Deveno
Science Advisor
906
6
indeed, it all depends on how you define "unitary", by inner product properties, or matrix properties. there is a deep connection betwen inner products and matrix multiplication.
 

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