Singular Value Decomposition - What is it used for, how do you apply it to practical problems?
813865
The singular value decomposition for a matrix A writes A as a product (hanger)(stretcher)(aligner). \n \n\nIt's an amazing and useful fact that every m x n matrix has a singular value decomposition. \n\n-------------------\nIn linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics.\n\nThe SVD can be seen as a generalization of the spectral theorem to arbitrary, not necessarily square, matrices. The spectral theorem says that normal matrices can be unitarily diagonalized using a basis of eigenvectors.\n\n---------------------------------\nSingular value decomposition takes a rectangular matrix of gene expression data (defined as A, where A is a n x p matrix) in which the n rows represents the genes, and the p columns represents the experimental conditions.\n\n---------------------------------\nA very powerful set of techniques dealing with sets of equations or matrices that are either singular or numerically very close to singular is the so-called singular value decomposition (SVD). SVD allows one to diagnose the problems in a given matrix and provides numerical answer as well.\n\nAny m x n matrix a (m >= n) can be written as the product of an m x n column-orthogonal matrix u, an n x n diagonal matrix with positive or zero elements, and the transpose of an n x n orthogonal matrix v:\n\n\nI hope that all these are useful, good luck!!!!!!!!!!!!
1018028
You need to know what this is equivalent to in order to form a eigenvalue problem.  And this is far different from finding the arbitrary roots to the quintic.  Check link for quick primer.
1178962
To solve this it's a pretty straightforward application of Laplace transforms. You can do this question using convolutions as follows:\n\nFirst take the laplace transfom of the ODE:\n\nlet L{y(t)}(s) = Y(s) be the laplace transform of y(t).  Then we have:\ns^2(Y(s))  - 2s + 3Y(s) = 1/s^2 - (e^-s)/s\n\nrearranging we have Y(s) = (1/s^2 - e^-s/s + 2s) (1/(s^2+3)) = G(s) H(s) where G(s) = (1/s^2 - (e^-s)/s)  and H(s) =  (1/(s^2+3)) \n\nnow using the inverse laplace formula for the convolution of two functions:\n\ny(t) = L(^-1){Y(s)} = integral from 0 to t of (h(t-x)g(x)dx) + 2cos(sqrt(3)t)\n\n where h(t) = 1/sqrt(3) sin(sqrt(3)t) and g(t) = (t - u1(t))\n\nand the integral can be evaluated. \n\nSorry for the readability but I don't think yahoo answers can render math text properly. Also, as a note, this question would be far easier to solve using a partial fraction method rather than unecesarily using convolutions.
897839
The angular change ratio from the planar field
849482
Have you thought about using FFT (Fast Fourier Transform)? That should be easy to implement.
839609
Curl, divergence, gradient.
320722
The central limit theorem still applies.\n\nWhen variables are correlated a principal component analysis will find the independent variables for which the CLT applies.
282293
The techniques used to solve systems of linear equations (substitution and elimination) tend to do a pretty good job of solving quadratic systems.\n\nThis page does a pretty good job of demonstrating how:
82021
I wish your question was stated more clearly. Do you have sound data? From a microphone? \n\nOne of the things most often done is to estimate the spectral content of the data. If ou are inexperienced, which you appear to be, I would start with a standard direct periodogram estimate. \n\nSubtract the mean from your data. Take a discrete Fourier transform of the remaining data (it is easily done with the FFT algorithm, if the data are sampled at uniform intervals). Plot the modulus squared of the result. \n\nThere is a tremendous amount of sophisticated analyses done on sound samples. It is way beyond the scope of yahoo answers to be able to give much detail. You should do some research and reading in the library and on the web if you want more.
1069671
yep...scalars multiply vectors component by component just like you have shown.  you can also show it by plotting out the original vectors, scaling them appropriately, adding them, and finding the resukltant components.
790217
Great question.\n\nMaybe you can work backwards from the characteristic polynomial.  You know the roots so:\n\np(l) = (l+2)(l-1)(l-3)  right?  Now multiply out, the coeff of the linear term is the negative of the trace.  The sum of the eigenvalues is equal to the trace.  The constant term is the determinant.  \n\nor, start with the diagonal matrix and apply a basis transformation to get a non-triangular matrix.  \n\nor, choose 3 orthogonal vectors to call the eigenvectors, then find the transformation of the ....(out of steam)
693668
Fourier analysis is a mathematical technique that has application in a lot of fields.  It's often used for retrieving meaningful data from statistical "noise", for instance..  it also shows up in antenna and filter design, signal analysis, and many other areas.
875558
click here:http://en.wikipedia.org/wiki/Gaussian_elimination
925102
Discriminant is part of the solution.  Just considering it tells you the types of solutions that you have.  \n\nThere are three possibilities:\n1. 2 real roots,\n2. 2 complex roots,\n3. 1 repeated real root.
210386
I think the point may be to use synthetic division.  Look that up in your text.
694227
Both Laplace and Fourier transforms are used for transformation from time-domain to freuency domain.\n\nThe basic difference between them can be understood from the transformation equations, We see that the Laplace transform can be viewed as a generalization of the Fourier transform from the real line (a simple frequency axis) to the entire complex plane. We say that the Fourier transform is obtained by evaluating the Laplace transform along the jw axis in the complex s plane. \n\nAn advantage of the Laplace transform is the ability to transform signals which have no Fourier transform.The Laplace transform can be seen as the Fourier transform of an exponentially windowed input signal
1336
Besides large systems of equations, matrices are also useful to solve for efficient portfolios.  The relationships between various asset classes are grouped in a "covariance matrix".  The variance of the portfolio can be calculated using simple matrix multiplication of the covariance matrix and the allocation to the portfolio.\n\nAlso, matrices are used in optimization problems, and regression analysis.