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Is it true that the transition matrix between two orthonormal bases is orthogonal?

If true, why? I do really not understand. How can I prove that (some ideas)? Thank you all!

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    $\begingroup$ Think of writing $v=Mu$ with $u$ having a single component equal to $1$, and all other equal to $0$. What is $u$ then? What is $v$? What do the columns of $M$ represent? What can you say of them as vectors? Think about it, and write an answer to your own question! $\endgroup$ – Anonymous Jan 24 '17 at 10:59
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    $\begingroup$ This is false in general. However, it's true if you are talking about orthonormal bases with respect to the real Euclidean inner product and real orthogonal matrix. $\endgroup$ – user1551 Jan 24 '17 at 11:15
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    $\begingroup$ Instead of jumping straight from the first orthonormal basis to the second, it may help to stop off at the standard basis in between. $\endgroup$ – G. H. Faust Jan 24 '17 at 11:18
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As user1551 points out, it's false in general, but true when talking about orthonormal bases with respect to the real Euclidean inner product and real entries.

So for easier understanding, let's suppose this is given and the vector space is $(\mathbb{R}^n, \langle . \rangle)$ with standard euclidean product, let $e = (e_1, \dots, e_n)$ be standard basis and have two orthonormal bases $b = (b_1, \dots, b_n)$ and $c = (c_1, \dots, c_n)$.
Have two orthogonal matrices $B$ and $C$, where the columns of $B$ (resp. $C$) are exactly the basis vectors of $b$ (resp. $c$) expressed in coordinates of the standard basis. (standard situation in e.g. an exercise, where the matrix columns are just the vectors)

Let $M(id, b, c)$ denote the change of base matrix from $b$ to $c$. It's always helpful to remind yourself of what an entry in a change of base matrix stands for. The $j$-th column if $M(id, b, c)$ is the vector $b_j$ expressed in coordinates of $c_1, \dots, c_n$

As you probably know, when changing basis from $b$ to $c$, you can also first go from $b$ to $e$ and then from $e$ to $c$, i.e.

$$M(id, b,c) = M(id, e, c) \cdot M(id, b, e)$$

What is $M(id, b, e)$? As said, the $j$-th column of this matrix is $b_j$ expressed in coordinates of the standard basis. So $M(id, b, e) = B$.

  • Can you say which matrix $M(id, e, c)$ is equal to?
  • Then, with the knowledge of $B$ and $C$ being orthogonal, it will be easy for you to show that $M(id, b, c)$ is orthogonal as well.
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