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The question is as follows: If A is a unitary matrix for a a linear transformation from V to V, where V is a real euclidean space and {v1,...,vn} is an orthonormal basis of V, then show that {Av1,...,Avn} is an orthonormal basis.

I know that if A is a unitary matrix that it can be diagonalized, i.e. there exists distinct linearly independent eigenvectors X1,...XN such that P (where P is the matrix made up of the columns of the eigenvectors) is invertible such that AP=PD for some diagonal matrix D.

I also know that the rows and the columns of unitary matrices are orthonormal.

How can I relate these facts to the question? Thanks

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  • $\begingroup$ If $A$ is unitary, what can we say about $\langle Av,Aw \rangle$ and its relation to $\langle v,w\rangle$? Can you translate the statement on rows and columns of $A$ to something which can be said about $A$ and $A^*$ (and $AA^*$?). $\endgroup$ – Roland Dec 21 '15 at 0:57
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$\langle A v_i, A v_j \rangle = \langle v_i, A^* A v_j \rangle = \langle v_i, I v_j \rangle = \langle v_i, v_j \rangle = 0$ for $i \neq j$ by the definition of adjoint.

Thus, $\{A v_i\}$ are a collection of orthogonal vectors. Since you have the same number of vectors as a basis and orthogonal implies linearly independent, this is a basis.

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