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Martin Sleziak
Martin Sleziak
07:08
in Mathematics, 15 mins ago, by maths student
Suppose A is orthogonal matrix and B is matrix similar to A. Then is it true that B is orthogonal?
I do not think this is the case.
The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is orthogonal and it is similar to $\begin{pmatrix} i & 0 \\ 0 &-i \end{pmatrix}$
This could by true if we consider matrix which is orthogonally similar to $A$, but I am not sure to which extent this is useful.
 
2 hours later…
maths student
maths student
08:48
@MartinSleziak the example you have given has underlying field is complex number, so my question is there any example where underlying field is real number?
Martin Sleziak
Martin Sleziak
@mathsstudent What about expressing some rotation in a basis where the vectors are not orthogonal to each other?
$$\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} -1 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix}$$
Did I make some mistakes there?

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