# Tag Info

16

You can always modify the matrix so that the most negative eigenvalue is also the one with the largest absolute value, and hence corresponds to the first in the list returned by EigenVectors. An upper bound for the largest absolute value of any eigenvalue is the Hilbert-Schmidt norm. So you can rescale your matrix by subtracting this norm times the unit ...

15

Here's my attempt. To get the matrix representing the Laplacian I use LaplacianFilter on an array of symbols and CoefficientArrays to extract the coefficients. n = 200; shape = ArrayPad[ConstantArray[0, {n/2, n/2}], {{0, n/2}, {0, n/2}}, 1]; shapeVector = Flatten @ Position[Flatten @ shape, 1]; symbolArray = Array[x, {n, n}]; symbolLaplacian = ...

12

While the other answers are nice, the icon deserves a closer look: Note, in particular, that four of the six edges are not constrained by the ostensible Dirichlet boundary conditions, nor is it clear that they solve a Neumann problem. And indeed, as I noted in the comments this is supported by the OP's first link. In short, to produce the logo, they took ...

12

With the understanding that the criterion for including a point in the 3D matrix is that it has 3 unique domains in any of the 26 points surrounding a zero value in the watershed here is a simple way to extract the data. In the case where zero values in the watershed are in minority it may be the fastest approach to getting a list of points that fulfill the ...

11

I had this laying around from a course in numerical linear algebra I taught a few years ago. Here's a matrix whose nonzero elements describe the basic shape. size = 50; nw = Partition[Table[i, {i, 1, size^2}], size]; sw = Partition[Table[i, {i, size^2 + 1, 2*size^2}], size]; se = Partition[Table[i, {i, 2*size^2 + 1, 3*size^2}], size]; L = ...

11

The phase (and length) of the eigenvectors is completely undetermined unless you specify extra conditions in addition to the eigenvalue equation. Given that you don't have any additional conditions, it's not surprising that there is no well-defined way to plot the real and imaginary parts of each eigenvector component. A simple condition that makes the ...

9

From the Documentation on Numerical Implementation: Machine-precision matrices are typically converted to a special internal representation for processing. SparseArray with rules involving patterns uses cylindrical algebraic decomposition to find connected array components. Sparse arrays are stored internally using compressed sparse row formats, ...

7

The symbol $\epsilon$ is called the Levi-Civita tensor, it is usually written in terms of its components $\epsilon_{i j k}$. Having defined two vectors (i.e. first order contravariant tensors): a = { a1, a2, a3}; b = { b1, b2, b3}; we have to build up a tensor product of two vectors and Levi-Civita tensor, i.e. we get a new tensor of valence $(3,2)$ ...

7

s = {x, y} /. Solve[a x + y == 7 && b x - y == 1, {x, y}][[1]] {8/(a + b), -((a - 7 b)/(a + b))} lsa = LinearSolve[{{a, 1}, {b, -1}}, {7, 1}] {8/(a + b), (-a + 7 b)/(a + b)} f = LinearSolve[{{a, 1}, {b, -1}}]; lsb = f[{7, 1}] // Simplify {8/(a + b), -((a - 7 b)/(a + b))} s == lsa == lsb // Simplify True Solve can handle a ...

7

The order of eigenvalues is the most convenient order for the algorithm, which find these eigenvalues. You can always order them as you want very simply a = # + #\[Transpose] &@RandomReal[1, {10, 10}]; {ε, ψ} = Eigensystem[a]; {ε, ψ} = {ε[[#]], ψ[[#]]} &@ Ordering[ε]; Furthermore, the eigenvalues can be complex for non-Hermitian matrices. There is ...

7

You have to be a bit careful here; your last approach does not give the desired matrix F: (LinearSolve[dV, dv] // Transpose).dV === dv False Then what does give the correct output? We can use the fact that for generic matrices $A$ and $B$ we have $A.B = (B^T.A^T)^T$ and write F = Transpose @ LinearSolve[Transpose @ dV, Transpose @ dv] And indeed, ...

6

I'll make this a response since it seems to be a popular topic today (it showed up independently on the Wolfram Community forum). One can usually do this by finding the eigenvectors for a random linear combination of the matrices. This works for nonderogatory matrices, that is, ones that do not have nontrivial blocks in their Jordan decomposition. In ...

6

Unfortunately the various Solve like functions don't show the same behavior concerning string-type variable names. Some do accept strings (e.g. NDSolve in version 9): NDSolve[{"x"'["t"] == 0.1*"x"["t"], "x"[0] == 1}, "x", {"t", 0, 1}] but others don't. For your case, Eliminate obviously doesn't, but you only need to convert those variables which you want ...

6

x = {3, 1} y = {2, 5} a Defer[x] + b Defer[y] /. First@Solve[a x + b y == {7, 11}, {a, b}] (* x + 2 y *) Note that the output is usable as such: evaluate it and you'll get the combination result. If only integer coefficients are desired, changing the Solve to something like: Solve[a x + b y == {7, 11} && {a, b} \[Element] Integers, {a, b}] ...

6

The compatibility information at Compatibility/tutorial/LinearAlgebra/MatrixManipulation says These functions were available in previous versions of Mathematica and are now available on the web at library.wolfram.com/infocenter/MathSource/6770: LinearEquationsToMatrices InverseMatrixNorm ConditionNumber You can download the original package ...

5

The set of equations given by $$e_p = \sum_{i=1}^D c_i\ p^i,\quad p\in\{1,\ldots,D\}$$ where the $e_p$ are the diagonal elements (or eigenvalues) of $\mathbf{E}$ can be written as $$\mathbf{e} = \mathbf{c}\ \left( \begin{array}{ccc} 1 & 1 & \ldots & 1^{D-1} \\ 1 & 2 & \ldots & 2^{D-1} \\ \vdots & & \ddots & \vdots ... 5 You could define a function that constructs the product of Pauli matrices as follows. I use KroneckerProduct here because you are planning to form the matrix product with a 2L\times2L matrix, so we have to have the Pauli matrices arranged in a corresponding block matrix: pauliProduct[n_] := Module[{l = Length[n]}, Total@MapIndexed[ KroneckerProduct[ ... 5 The two eigenvalues are degenerate. In general whenever you have degenerate eigenvalues there is arbitrariness in how to select the eigenvectors. You can construct a constant vector from the two degenerate eigenvectors like so: m = Import["https://dl.dropboxusercontent.com/u/63413473/ExampleMatrix.mx"]; {evals, evecs} = Eigensystem[m, -2]; You can check ... 5 The problem here can arise because of numerical underflow which appears for sufficiently large dimension of the problem. Some numerically very small number multiplies the parameter "a" and therefore "a" does not appear in the "solution". Consider a simple example Define the matrix m (fill it with random numbers, here exponentially distributed) In[263]:= ... 5 Dense blocks One can split matrices by blocks and use these blocks as a dense matrices blockSize = 100; m1[a_] m2[b_] ^:= a.b; part = DeveloperPartitionMap[If[Length@#@"NonzeroValues" > 0, m1@Normal@#, 0] &, a, {blockSize, blockSize}]; a2.u_ ^:= Flatten[DeveloperToPackedArray[ part.Developer`PartitionMap[m2, u, blockSize]], 1] I have ... 5 Ok, I'll go ahead and answer my own question. After some thinking I came up with this: getIntersection[l1_, l2_] := Module[{n, ker, coeffs}, ker = NullSpace[Transpose[Join[l1,l2]]]; n = Length[l1]; coeffs = Map[Function[v, v[[1 ;; n]]], ker]; Return [Map[Function[v, v.l1], coeffs]]; ]; Can this be made any faster? 5 Maybe MapThread gives the levelspec control you need: lst = {{{a, b}, {c, d}}, {{w, x}, {y, z}}}; g @@ MapThread[f, lst, 1] (* or just g @@ MapThread[f, lst] -- thanks: Mr.W *) (* g[f[{a,b},{w,x}],f[{c,d},{y,z}]] *) g @@ MapThread[f, lst, 2] (* g[{f[a,w],f[b,x]},{f[c,y],f[d,z]}] *) or, g @@ Thread[f @@ lst] (* g[f[{a,b},{w,x}],f[{c,d},{y,z}]] *) or ... 4 note see bottom of answer for a required fix for version 10+ A 2D example of what i suggested in my comment: Based on Belisarius example..we get three lines obviously not connected properly: m = SparseArray[{i_, j_} -> Sin[i j 9 /10 y], {3, 3}]; alle = Table[Eigenvalues[m], {y, 0, 1, .1}]; original = Show[ MapIndexed[ListPlot[Flatten[Take[alle, ... 4 Thread is the inappropriate tool here as it does not hold its arguments, i.e. it does not have the attribute HoldAll: Attributes@Thread (* {Protected} *) So, ConjugateTranspose[v].H.v evaluates before Thread can operate on it. Also, Thread will attempt to thread across H, as well, which is clearly not what you want. The correct tool to use here is Map: ... 4 Ordering[Norm /@ Last @ N[Eigensystem[system]]]; gives you the ordering by norm. You can apply this on your eigenvalues and eigenvectors, e.g. Eigenvectors[system][[%]] EDIT To apply this on a list of matrices: (# &@Ordering[Norm /@ N[#]]) & /@ Eigenvectors[#] & /@ {mat1,mat2,...,matn} 4 This is a problem known as finding 'moments of moments'. Notation Define the power sum s_r:$$s_r=\sum _{i=1}^n X_i^r$$Your problem only involves s_1. The Problem Let \left(X_1,\ldots,X_n\right) denote a random sample of size n from a population random variable X. The problem is to find:$$ E\Big [\Big (\frac1n\sum_{i=1}^n X_i\Big)^2\Big ...

4

Manipulate[Row[{ Plot3D[2 x +3y + 3 x^2 +4 y^2 +5 x y,{x,0,100},{y,0,100}, BoxRatios->1,ImageSize->400, MeshFunctions->{#3&},Mesh->{{k}},MeshStyle->Directive[{Thick,Red}]], ContourPlot[2 x +3y + 3 x^2 +4 y^2 +5 x y==k,{x,0,100},{y,0,100}, ContourStyle->Directive[{Thick,Red}],ImageSize->400, ...

4

The LinearAlgebra package has been deprecated since Mathematica 5, and is no longer bundled with Mathematica 9 or newer. You can still download a part of it (which contains the MatrixConditionNumber function) at the URL that jtbandes gave in his answer. First, we need to load the package: (* Be sure to install the above linked package in a "LinearAlgebra" ...

4

This is a bit long for a comment. You can use a tandem of LinearSolve and NullSpace. But for exact problems this will, I'm fairly sure, use dense matrices. That takes you back to what Solve is doing anyway, under the hood. Assuming your inputs are integer or rationals you might avoid dense matrix algebra as follows. Use numerical methods to find a single ...

4

Update: after clarification I propose f[m_] := Block[{\$Assumptions = Alternatives @@ Flatten@m ∈ Reals}, Conjugate[m[[1, 1]]] // Simplify] f[{{x, x^2}, {h[x], S[y]^2}}] (* x *) Previous test-based answer: From the documentation of MatrixQ Test if a matrix has real numeric entries: MatrixQ[{{Pi, Sin[1]}, {Cos[2], E}}, Im[#] == 0 &] ...

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