This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.
Let
and
be orthogonal
matrices. Determine which of the following are orthogonal (provide reasons for answers).
i)
(my answer is below, but I'm not sure if it is concrete enough)
Let
. Since
is orthogonal, then
. Therefore,
is orthogonal.
ii)
(I'm afraid I'm not sure about this one)
If
is also an orthogonal
matrix, must
be orthogonal?
I've inferred that
is orthogonal, using the following three equivalent statements for some
matrix
:
a.
is orthogonal.
b.
for every
c.
for every
The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).
Can anyone help me with this?