What is a good way to explain the concepts of eigenvalues and eigenvectors?
547857
A matrix represents a way of rotating and stretching some n-dimensional space in a linear way. An eigenvector is something that is just stretched or reflected, not rotated. The eigenvalue is the degree of stretch and (if negative) reflection. So for example, a rotation in 3 dimensions always has an axis. A vector in the direction of the axis will be an eignevector and the eigenvalue will be 1 since the vector stays the same length.
199628
Assuming, of course, that each of the vectors has 3 entries...\n\nDo you know how to put a matrix formed from those four vectors in row-echelon form?\n\nI'm trying to decide the best way to teach this to you, which is why I need to know if you know how to get the matrices in REF, but the final "Answer" is to see if three of those vectors are linearly independent. If three of them are linearly independent, then they must span R^3.\n\nFor your second question, no. Remember that a singular matrix is a square matrix that is not invertible. Therefore, a singular matrix will not have a pivot in every column, which means that the columns are linearly dependent, which means that the rank is less than the dimension. Any square matrix with a column that is all 0's will will be singular.
173113
Below are some links with some cool diagrams.  I'm not sure if it's specifically what you wanted though.  The second link has some really cool information about spectral transitions in a LASER.
199213
You probably mean eigen-values. We say that x is an eigen value for the square matrix A if there is a vector v such that Av=xv. Eigen-values are usually found by solving det(A-xI)=0\nwhere I is the identity matrix.
787521
Example:\nIf z=f(u,v),\nu=g(x,y) and\nv=h(x,y), then\n(all derivatives are supposed to be partial derivatives)\ndz/dx=dz/du*du/dx+dz/dv*dv/dx\ndz/dy=dz/du*du/dy+dz/dv*dv/dy\n\nAlternatively, you can set of the matrices\n[[dz/du dz/dv]]\nand\n[[ du/dx du/dy]\n[dv/dx dv/dy]]\nand get that\n[[dz/dx dz/dy]]\nis the product of the two matrices.
1090160
E-mail me if you want a detailed step-by-step explanation. Illustrations extra..... :)
159293
Depends on what you are analyzing.  Matrix algebra is used extensively in communications and signal processing applications.  Not very much, though, in financial applications.
830322
I just noticed that two rows are the same, right?  This is big, but I'll have to study up what it means.  Intuitively, I think that it means that the matrix has a null space, and thus and eigenvalue of zero.  \n\nor is it a type?\n\n\nRead up on it:\n\nhttp://en.wikipedia.org/wiki/Jordan_normal_form
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)
238922
Howstuffworks has good diagrams and explanations that may help you.  Good luck.
771697
Try Google or Yahoo.  You'll find worked examples there.  I think it needs diagrams to understand; they and the accompanying explanations are too complex to attempt to reproduce here.
773251
The Wikipedia entry discusses the linear algebraic proof of the existence of the SVD factorization (if that is what you want)\n\nhttp://en.wikipedia.org/wiki/Singular_value_decomposition\n\nA few other easy-to-understand links:\n\nFrom netlib:\nhttp://www.netlib.org/lapack/lug/node53.html\n\nFrom Numerical Recipes:\nhttp://www.nrbook.com/a/bookfpdf/f2-6.pdf
747794
Okay, it's "beginning algebra", not "begging algebra".  For a good start, check out "Elmentary Algebra" in Wikipedia, and check out its external links.
910755
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on two parameters. Matrices can be added, multiplied, and decomposed in various ways, making them a key concept in linear algebra and matrix theory.
742808
It's called spectral analysis and you do this type of analysis on an operator on a Hilbert space. It is a generalization of the study of eigenvalues and eigenvectors of a matrix. If you are at a level where this question could come up, you should do the work yourself.
1021968
http://en.wikipedia.org/wiki/Eigenstate
1130286
There is a primer here:\nhttp://www.phon.ucl.ac.uk/courses/spsci/matlab/primer40.pdf\n\nand an online course here:\nhttp://www.phon.ucl.ac.uk/courses/spsci/matlab/\n\nI may just be reviewing them myself!