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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