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 Post subject: Relation of an inner product to an SVDPosted: Thu, 5 Apr 2012 12:57:15 UTC
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On the physicsforums.com forum, the poster sunjin09 pointed out a relation between an inner product and a singular value decomposition of a product of matrices.

Let be a by dimensional matrix.
Let be a by dimensional matrix.
Let be a dimensional row vector.
Let so is the dimensional column vector obtain by interpreting as defining a linear combination of the columns of .
Let be an dimensional row vector.
Let so is the dimensional column vector obtained by interpreting as defining a linear combination of the columns of .

Use to indicate the transpose of a matrix . Assume A,D,a,b are real valued.

sunjin09 points out that the inner product can be written in terms of the SVD of .

Let be the singular value decomposition of

So the value of the inner product is determined by the and the SVD of . (Which, of course, implies that you if you kept and constant while varying and , the inner product would remain constant.)

Can this fact be understood in terms of any of the intuitive ways of looking at the SVD? I've read descriptions of how the SVD can be interpreted as the natural way to express a single matrix as mapping but I don't see how to apply that to an understanding of this result.

In the thread where this result arose, and represent two sets of dimensional column vectors and the span of the column vectors in only intersects the span of the column vectors of at the zero vector. Sunjin09 thinks that of is the largest singular value in the matrix .

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 Post subject: Re: Relation of an inner product to an SVDPosted: Thu, 5 Apr 2012 16:40:25 UTC
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http://en.wikipedia.org/wiki/Singular_v ... omposition

explains the theory pretty well I think.

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 Post subject: Re: Relation of an inner product to an SVDPosted: Thu, 5 Apr 2012 18:36:13 UTC
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Is sunjin09's result in that article?

I'm familiar with the theory of the SVD and I've looked at that article. I don't see how to apply the interpretations of the SVD to explain sunjin09's result - I meant to explain it in some intuitive manner. The algebraic explanation of his result is clear.

Perhaps what I don't understand is the significance as a mapping being related to the inner product of two vectors from the respective column spans of and . Is this some property of linear operators and inner products that "everybody knows"?

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