Closest Unitary Matrix to a given matrix

Speaker: Poornendu Kumar

Date: 11th-September- 2018

Time: 9:15pm-10:15pm

Abstract: 

In various physics and chemistry experiments we get the measurements as vectors. And, theoretical considerations often suggest that these vectors should be orthogonal/orthonormal. One way to achieve this is to apply Gram orthogonalisation to obtain a new set of orthogonal vectors. But, often the orthogonal set of vectors obtained via Gram process is not close to the original set of vectors. This led to the problem of ‘orthonormalising’ a set of vectors without changing the vectors much. Put more precisely, we can see the set of vectors as a matrix (square) A and we may ask for the unitary matrix U which is closest to A (in operator norm sense).

Interestingly, the solution of this problem goes through the polar decomposition of the matrix A, which gives an answer but conceals much of the geometry. In this talk, I will present a proof of the fact that the closest unitary matrix to given matrix A is the one which is obtained in it’s polar decomposition. There are natural generalizations of this problem which I shall discuss if time permits.

 

Prerequisites: 

Basic Linear Algebra

(Knowing the statement of polar decomposition would be enough.)

 

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