What is the best way of introducing singular value decomposition (SVD) on a linear algebra course? Why is it so important? Are there any applications which have a real impact?
it's a natural generalization of the spectral theorem you asked about in your last post. for the truly mathematically-inclined, this is motivation enough.
but, it can also be viewed the following ways:
it allows us to compute "the best possible orthogonal bases" of the domain and co-domain of a linear transformation of finite-dimensional linear spaces, in this sense that the matrix for T in these bases is as "simple as possible" (diagonal).
geometrically, this allows us to view any linear transformation as:
rotation+scaling map+rotation.
one way to see this is to "follow what happens to a unit n-sphere" (under the norm induced by the inner product we are using), for each of the three linear transformations in the decomposition.
it allows us to calculate the pseudo-inverse of a matrix, which is used in solving "least squares" (best fit) solutions such as finding the best fit polynomial of a given degree that matches the data (the polynomial isn't linear in its "indeterminate" variable, but IS a linear function of its coefficients).
in signal processing, the size of the singular values of a matrix are related to "which signals carry information" and "which signals are noise". calculating the SVD allows for "better (noise) filter design".
variations of the SVD are used in such diverse applications as: optical character recognition, radar target recognition profiles, 3d reconstruction from 2d images, fingerprint analysis, and weather prediction.
in general, calculation with a given mxn matrix is hard, evaluating the image of a given domain vector requires mn2+m numerical operations. if m is near n, this is O(n3) operations. using the SVD reduces this to O(n) operations (with, of course, an "up-front cost" of calculating the unitary matrices used in the decomposition). if one is going to use a particular linear transformation several times, this is well worth the effort. as the great mathematican indiana jones said: "choose (your bases) wisely".
I asked online questions about Proposition 2.1.1:
The answer I got is the following:
I have some questions about the answer I got.
When the person answering says:
##1.##
Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##?
But I don't understand what the author meant for the rest of the sentence in mathematical notation:
##2.##
In the next statement where the author says:
How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho.
In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states
"Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels).
Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product
$$
\langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$
where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...