Suppose V is a n-dimensional linear vector space. {s_1,s_2,...,s_n} and {e_1,e_2,...,e_n} are two sets of orthonormal basis with basis transformation matrix U such that e_i=Sum_j U_(ij) s_j.

Krish Schmitt

Krish Schmitt

Answered question

2022-09-30

Suppose V is a n-dimensional linear vector space. { s 1 , s 2 , . . . , s n } and { e 1 , e 2 , . . . , e n } are two sets of orthonormal basis with basis transformation matrix U such that e i = j U i j s j .
Now consider the n 2 dim vector space V V (kronecker product) with equivalent basis sets { s 1 s 1 , s 1 s 2 , . . . , s n s n } and { e 1 e 1 , e 1 e 2 , . . . , e n e n }. Now can we find the basis transformation matrix for this in terms of U?

Answer & Explanation

Paige Paul

Paige Paul

Beginner2022-10-01Added 11 answers

Simply use bilinearity of the tensor product. Since, as you said, e i = j U i j s j , we can write any basis vector of V V as
e i e k = ( j U i j s j ) ( U k s ) = k , U i j U k s j s

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