Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A.(adsbygoogle = window.adsbygoogle || []).push({});

Now, the first (old) basis, in terms of the second(new) basis, is given by, say,

$$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$

How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for some unitary matrix, ##U##?

What is the relation between the matrix formed by the numbers ##S_{ij}## and ##U##?

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# I Change of Basis and Unitary Transformations

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