Can you describe what an eigenvalue/vector is in plain English?
545935
http://en.wikipedia.org/wiki/Eigenvalue\n\nI read through this and it made more sense to me this time than it ever has.  You can think of the solutions to a problem as vectors that point in a certain specific direction.  An eigenvector is a solution vector that points along the same direction but has a different scale length compared to the original vector.  That scale length is the eigenvalue/s.
856009
false,\nuse the identity and -identity,\nboth are invertible, but the sum is the zero matrix, that is NOT invertible.....
145647
In general it used in negetive sense.\nFor example you are supposed to implement Matrix multiplication( or built a matrix object).\nThere is two way to show that your implementation works,\n1. create some matrix in your code and show with that. to test for another value you have to change the code.\n2. Take matrix as input from user.\n\nFirst practice is known as hardcore progrmming.
331428
Can you show that T has a real eigenvalue? If so, let v be an eigenvector corresponding to this eigenvalue. Then, the line tv for all t in Reals is invariant under T :)
102340
I assume you mean invertible, not convertible.\n\nIt depends on how detailed you want to get. If you do not want to dig in extremely deeply, you can just do it by construction. \n\n1. A symmetric matrix M has an orthogonal decomposition\n\nM=O'DO\n\nwhere O is an orthogonal matrix, O'O=I, the identity, and D is a diagonal matrix with eigenvalues of M down the diagonal.\n\n2. If M is positive definite, then the diagonal elements of D are >0\n\n3. Produce the matrix \n\nA=O'(D^(-1))O\n\n\n4. Verify that A is the inverse of M\n\ni.e., \n\nA*M=M*A=I
583382
Try the moebius spiral. It's pretty interesting because it's non orientable. \n\nIt's 3d (2 parameters s and t) and described by the parametrized vector components:\n\n[3*cos(s)+t*sin(s/2),\n3*sin(s),\nt*cos(s/2)]\n\n(every component of the vector in a new line)\n\nThe parameter ranges are:\n\ns=-2*Pi...2*Pi,t=-1..1\n\nMAPLE INSTRUCTIONS TO PLOT IT:\n\nwith(plots);\nplot3d([3*cos(s)+t*sin(s/2),3*sin(s),t*cos(s/2)],s=-2*Pi...2*Pi,t=-1..1);\n\nSorry, that instruction is big and not displayed well, but I guess since you know the parametrized vector you can find it.\n\nWHAT DO YOU WANT IT FOR? WANT TO PUBLISH ANYTHING?
307198
It is linear on intervals, on (- infinite, 0) and on (0, +infinite) but not on the whole R. \n\nTry plotting it, you get a reversed V, not a straight line.
851884
They are unitary vectors in that their magnitude = 1 so that i dot j = i dot k =j dot k =1 . one being unity.\n\nThey do not have _units_ like displacement, velocity, magnetic field strength associated with them as they are designed to provide a general standardised framework for vector analysis where the vector quantities manipulated may be any sort of physical vector property.
945015
Modal analysis is one application of Eigenvalue analysis but Eigenvalue analysis can be used for other things.\n\nIn the context of modal analysis (using FEM), the eigenvalue=eigenfrequency=natural frequency (mode).\n\nThe eigenvector is the diplacement vector or mode shape of the mode.\n\nI haven't seen the term eigensystem, but it probably refers to the matrix and vector for the problem (i.e. everything as a whole).
440750
Density Matrix.\n\ndiagonal part + nondiagonal part or\nstate probabilities + transition probabilities.
333863
Diagonalisation is the answer here.  Not sure how this didn't turn up in a web search, as I found plenty of resources for it, but the idea is that you find a matrix that diagonalises your initial matrix, and this allows powers to be done trivially.\n\nAssuming you're aiming to get a matrix A to the power n:\n\nFind P such that:\nP^-1 A P = D, where D is diagonal.  Then:\nA = (P D P^-1) and:\nA^n = (P D P^-1)(P D P^-1)(P D P^-1)....\n\nThe P^-1 P terms cancel and you're left with:\nA^n = P D^n P^-1\n\n... and this is much easier to calculate as D^n is trivial :)\n\nA good reference is in the sources.  Hope that helps!
268402
Consider the matrix of the real parts of a Hermitian matrix as A and the matrix of the imaginary parts as B. Use the definition of Hermitian.\nAnother hint: If a_ij = a+ib, what would a_ji have to be if the matrix A is Hermitian ? What is then true about the diagonal of a Hermitian matrix?
700308
Seems more like a vector analysis question to me.
599681
It depends on exactly what you mean by a matrix. If you're talking about multi-dimensional or nested arrays in ActionScript, then here is a link to get you started...
1106173
The direction (angle) of |u| and |v|?
329083
When dealing with simple shapes, an area is spanned up by two vectors. The area shape will have a normal vector indicating how it is oriented in space, resulting from the cross product of the two linear vectors with the length indicating the area units.
978796
Study of data with respect to different spectral responses