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16

Using the properties of Block matrices: $$\det\begin{pmatrix}\mathbf A&\mathbf B\\\mathbf C&\mathbf D\end{pmatrix}=\det(\mathbf A)\det\left(\mathbf D-\mathbf C\mathbf A^{-1}\mathbf B\right)$$ To visualize your matrix: mat1 = mat; {mat1[[;; 10, ;; 10]], mat1[[;; 10, 11 ;;]], mat1[[11 ;;, ;; 10]], mat1[[11 ;;, 11 ;;]]} = Range@4; (* cool :) *) ...


11

The Pitsianis-Van Loan algorithm turns out to be surprisingly easy to implement in Mathematica: nearestKroneckerProduct[mat_?MatrixQ, dim1_?VectorQ, dim2_?VectorQ] /; TrueQ[Dimensions[mat] == dim1 dim2] := Module[{bv, cv, sig}, {bv, sig, cv} = SingularValueDecomposition[Flatten[ Map[Composition[Flatten, Transpose], ...


9

There is another option, using the relatively new tensor capabilities of Mathematica. This is pretty much copied from another answer by jose, but I don't need any assumptions here: TensorExpand[KroneckerProduct[X, X] + KroneckerProduct[-X, X]] (* ==> 0 *) TensorExpand[KroneckerProduct[2 X, 3 Y]] (* ==> 6 KroneckerProduct[X, Y] *) There is a ...


8

I needed this decomposition to answer another question, so I broke down and implemented it myself. The code is more or less a straightforward translation of the pseudocode in Golub/Van Loan: LDLT[mat_?SymmetricMatrixQ] := Module[{n = Length[mat], mt = mat, v, w}, Do[ If[j > 1, w = mt[[j, ;; j - 1]]; v = ...


7

$MachinePrecision is different from MachinePrecision. The former calls for an arbitrary precision calcluation, done at the same precision as the machine-precision one. The main reason one would want to use this is to enable precision tracking, which is absent for a true machine-precision calculation using MachinePrecision. And, there is your answer. ...


6

As @Guess who it is. states in the comments, an overdetermined linear problem can be solved using Mathematica's LeastSquares[] functionality. To input your above system of equations: a = {{1, -2, 2, 2, 0, -2}, {1, -2, 2, -2, 0, 2}, {1, -2, -2, -2, -4, -2}, {1, 2, 2, -2, -4, -2}, {1, 2, -2, -2, 0, 2}, {1, 2, -2, 2, 0, -2}, {1, -2, -2, 2, 4, ...


5

We will go by solving one equation at a time and generating the corresponding replacement rules. Beware of possible numerical instabilities. The following is the equivalence between the code in your edited example and my code on the previous incarnation of this answer. I think this is enough for you to use it. Please note that the only claim on the ...


5

Here's my relatively compact implementation of Glynn's formula, which incorporates the Gray code optimization: SetAttributes[GrayCode, Listable]; GrayCode[n_Integer] := BitXor[n, BitShiftRight[n]] permanent[mat_?MatrixQ] /; Equal @@ Dimensions[mat] := Module[{b = 2^(Length[mat] - 1)}, PadRight[{}, b, {1, -1}].(Times @@@ ...


4

Using as basis the great resource for core numerical algorithms below, I managed to implement a compiled linear solve which doesn't call MainEvaluate (so quite fast). I needed a linear solve for an optimization where the objective function requires inverting matrices, I was hesitating to use C++, but I preferred to stay in Mathematica. Resources ...


4

If you call Orthogonalize at the end, you're orthogonalizing the eigenvectors in a different order (i.e. after sorting on eigenvalue, rather than before). Orthogonalizing the same list in a different order usually gives a different output. Orthogonalize[{{1., 2}, {1, 3}}] (* {{0.447214, 0.894427}, {-0.894427, 0.447214}} *) Orthogonalize[{{1, 3}, {1., 2}}] ...


3

How about: av = Array[Subscript[a, ##] &, {2}]; bv = Array[Subscript[b, ##] &, {2}]; KroneckerProduct[av, bv] + KroneckerProduct[-av, bv] {{0, 0}, {0, 0}}


3

This is a somewhat high-brow way of showing the Cayley-Hamilton theorem, through the power of holomorphic functional calculus. As I mentioned in this answer, one of the standard ways to define a matrix function is through a Cauchy-like construction: $$f(\mathbf A) = \frac{1}{2\pi i} \oint_\gamma f(z)\, (z \mathbf I- \mathbf A)^{-1}\,\mathrm dz$$ where the ...


3

Here is a slightly compacted reformulation of belisarius's answer: a = Take[mat, 10, 10]; b = Take[mat, 10, -15]; c = Take[mat, -15, 10]; d = Take[mat, -15, -15]; rr = ArrayRules[d - c.SparseArray[LinearSolve[a, b]]]; detr = Det[SparseArray[rr /. (pos_ /; VectorQ[pos, IntegerQ] -> expr_) :> (pos -> C @@ pos), ...


2

We can set up your problem as follows, including the requirement that all vector elements be rational and non-zero: matrix = Table[Indexed[a, {i, j}], {i, 1, 6}, {j, 1, 6}]; instance = FindInstance[ Flatten@{ Equal @@@ Table[{matrix[[i, 1]]^2, Total[matrix[[i, 2 ;;]]^2]}, {i, 1, 6, 1}], Table[Indexed[a, {i, j}] != 0, {i, 1, 6}, {j, 1, 6}] ...


2

I write this answer with the caveat that I don't have Mathematica version 9 or later (the versions which now have this very belated function built-in), but with said caveat being offset by knowing a thing or two about the function of a matrix. ;) I'd have to agree with george's take that the docs for MatrixFunction[] could probably have explained things a ...


2

Try this: (Normalize /@ evecs // Transpose // Assuming[0 < \[Theta] < Pi, FullSimplify[TrigToExp@#]] &) /. {(1 + Abs[Cot[\[Theta]/2]]^2)^(-1/2) -> Abs[Sin[\[Theta]/2]]} (* {{-I Cos[\[Theta]/2], I Sin[\[Theta]/2]}, {Sin[\[Theta]/2], Cos[\[Theta]/2]}} *) Edit: if theta<0, (Normalize /@ evecs // Transpose // ...


1

Second update -- I should state in simplest terms the issue the OP is facing. The set-up. The equations for a given a are eqs = Table[Power[Sqrt[λ2/(μ2 c2)], n] p[0, n + 1] == β[n] p[0, 0] + Sum[α[n, k] p[0, k], {k, 0, a}] /. NumVal, {n, 0, a - 1}]; All the variables involved in eqs are given by vars = Table[p[0, n], {n, 0, a}]; (* starts ...


1

Except in special cases, the problem seems overdetermined. Consider for simplicity m = {{m11, m12, m13}, {m21, m22, m23}, {m31, m32, m33}}; n = {{n11, n12, n13}, {n21, n22, n23}, {n31, n32, n33}}; t = {{t11, t12, t13}, {t21, t22, t23}, {t31, t32, t33}}; Then, the first equation becomes t.m - n.t == 0 unless t is singular. The solution is Solve[t.m - ...



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