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11 votes
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Raise a Matrix to Arbitrary Power

A result can be obtained by direct computation. It can be seen that the first row of $A_k$ is $$(A_k)_{1, j} = 1.$$ Then, the first row of $A_k^{\nu+1}$ is produced by taking partial sums as follows $$...
Pantelis Sopasakis's user avatar
8 votes
Accepted

Symmetric matrix is positive semidefinite

This is not true. Random counterexample: $$ A=\pmatrix{1&\frac34&0&\frac12\\ \frac34&1&\frac34&0\\ 0&\frac34&1&\frac12\\ \frac12&0&\frac12&1}. $$ $A$ ...
user1551's user avatar
  • 140k
8 votes

Numerically computing eigenvalues -- what is it useful for?

One of the most important and widely used applications of eigenvalues specifically as opposed to singular values comes from from dynamical systems. Consider a linear ODE $$\dot{x} = Ax,$$ where $A$ is ...
whpowell96's user avatar
  • 6,017
6 votes
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Eigenvalue of diagonally dominant matrices

Choose an $n \geq 3$, and consider the matrix $$A=\begin{bmatrix} 1& -1& 0& \cdots & 0 \\ 0 & 1 & -1 & 0 & \cdots \\ \cdots & & & &\\ -1 & 0&0&...
abacaba's user avatar
  • 8,685
6 votes
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Why is in Matlab exp(pi * sqrt(163)/3) - 640320 = -2.3283e-10

First, that expression is not an integer, but very close to an integer $640320$. In fact, as @Peter mentioned, this is related to some deep theory of modular $j$-functions and Heegner numbers, e.g. ...
Seewoo Lee's user avatar
  • 15.2k
5 votes
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What unique value does Cramer's rule offer?

Cramer‘s rule is used in a lot of proofs. You can use it to get the $(i,j)$-th entry of the matrix $A^{-1}$ (see adjugate matrix), it be used to prove the that every smooth $C^1$ diffeomorphism is ...
Eric's user avatar
  • 1,130
5 votes
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Prove that $1 \leq \|A^{-1}\| _2\|A-B\|_2$

Let $v$ be a nonzero unit vector in $\ker(B)$. Then $$ \|A^{-1}\|_2\|A-B\|_2 \ge\|I-A^{-1}B\|_2 =\sup_{\|u\|_2=1}\|(I-A^{-1}B)u\|_2 \ge\|(I-A^{-1}B)v\|_2 =\|v\|_2=1. $$
user1551's user avatar
  • 140k
5 votes
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Why is this matrix always symmetric?

An initial simplification helps. Absorb $\frac{h^2}{8}$ into $B$ and forget about it. Now note that since $B$ is symmetric, our given matrix is symmetric if and only if $X:=B(I-A^{-1}B)^{-1}$ is ...
ancient mathematician's user avatar
5 votes

Intersection of two ellipses at exactly 2 points

Assume that $A, B, k$ are given where $A, B$ are symmetric positive definite $2 \times 2 $ matrices, and $k \gt 0$. The solution of $ x^T A x = k $ is the ellipse $ x = v_1 \cos t + v_2 \sin t $ ...
c'est pas normale's user avatar
5 votes

Intersection of two ellipses at exactly 2 points

We can assume WLOG (up to division by a constant) that the first ellipse $(E)$ has equation $$X^TAX=1$$ Let $X^TBX=\ell$ be the second variable ellipse that we will call $(E_{\ell})$. Result : the ...
Jean Marie's user avatar
  • 83.1k
5 votes
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Why is easier to get inverse of mass matrix?

The claim that "Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix" is more of a slogan than a precise theorem. This answer gives ...
Korf's user avatar
  • 706
4 votes

Eigenvalue of diagonally dominant matrices

It isn’t clear whether the statement you want to prove/disprove is $$ \operatorname{Re}(\lambda)\ge|\operatorname{Im}(\lambda)| \ \text{ for all eigenvalues $\lambda$ of $A$}\tag{1} $$ or $$ \...
user1551's user avatar
  • 140k
4 votes
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Numerically computing eigenvalues -- what is it useful for?

The eigenvalues of partial differential operators describing mechanical or electromagnetic systems are related to the resonance frequencies. For example, the frequencies at which a drum or guitar or ...
Wolfgang Bangerth's user avatar
4 votes
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Is it possible to apply the matrix inverse $(H(\textrm{Id}+F^T F)^{-1}H^T)^{-1}$ to a vector without explicitly calculating the inverse?

Yes. You wish to compute $$x = (H(I + F^TF)^{-1}H^T)^{-1}y$$ for any $y$. You do this by solving the linear system $$\begin{bmatrix} I + F^TF & H^T \\ H & 0 \end{bmatrix} \begin{bmatrix} ...
Carl Christian's user avatar
3 votes

Finding the bounds of the missing values of a symmetric positive semidefinite matrix

The determinant of your matrix is $-(x-1)^2$. Since a positive semidefinite matrix must have nonnegative determinant, the only possible value is $x=1$. You still need to check that the matrix is ...
Robert Israel's user avatar
3 votes

LU factorization distribution over addition

The answer is almost certainly no. The key fact to notice is that your matrix $A = I + \alpha D$ is a rank $n$ perturbation of the matrix $\alpha D$. You cannot profit from applying the Sherman-...
Carl Christian's user avatar
3 votes
Accepted

Efficient algorithm of rank-one update of the Cholesky decomposition

Tensorflow uses the implementation recorded in Krause & Igel (2015). reference: Krause, O., & Igel, C. (2015, January). A more efficient rank-one covariance matrix update for evolution ...
Andy Season's user avatar
3 votes
Accepted

Symmetric matrix with integer values is positive semidefinite

Again this is not true. $$ \det\begin{pmatrix} 9&8&7&6\\ 8&9&0&0\\ 7&0&9&0\\ 6&0&0&9 \end{pmatrix}=-5508. $$ Also, if you replace the $0$s with $1$s the ...
user10354138's user avatar
  • 33.3k
3 votes
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Conjugate gradient descent when $A$ is non-singular but not symmetric

CG does not converge to a solution, e.g., for $$A=\begin{bmatrix}1&1\\0&1\end{bmatrix}\text{ and } b=\begin{bmatrix}1\\1\end{bmatrix}.$$ The solution is $x=[0, 1]^T$, but CG stagnates at $x_k=[...
Algebraic Pavel's user avatar
3 votes

Solving Frobenius Norm Inequality with Moore-Penrose Inverse

The nicer approach is to leverage things that you already know about the MP pseudoinverse. Here's an approach you might like. Let $x_1,\dots,x_m$ denote the columns of $X$ and $e_1,\dots,e_m$ the ...
Ben Grossmann's user avatar
3 votes
Accepted

Quadratic Convergence of Newton's Method for Matrix Inverse Calculation

Let $V$ be a normed space and let $\{x_n\}_{n=1}^\infty \subseteq V$ be convergent with limit $x$. We say that the convergence has order $p>1$ if there exists $c>0$ such that $$\frac{\|x-x_{n+...
Carl Christian's user avatar
3 votes
Accepted

Bounding the condition number by the eigenvalues

If $\lambda$ is an eigenvalue of $M$ and $v\ne0$ is a corresponding eigenvector, then you have that $$ \|M\| = \sup_{x\ne 0}\frac{\|Mx\|}{\|x\|} \ge \frac{\|M v\|}{\|v\|} = \frac{|\lambda| \|v\|}{\|v\|...
PierreCarre's user avatar
3 votes
Accepted

Why does A-sI have a better condition number than A?

The identity matrix is an optimally conditioned matrix: $κ(𝕀)=1$. Moreover, condition numbers are independent of rescaling: $κ(λA)=κ(A)$ for $λ≠0$. What $A-s𝕀$ does is it biases $A$ towards an ...
Hyperplane's user avatar
  • 11.8k
2 votes

How to find a sparse basis of the null space of a large sparse matrix using QR decomposition

The computation of sparse null bases is very active research due to their applications in the solution of saddle point problems, which occur often in large-scale applications with sparse matrices. ...
whpowell96's user avatar
  • 6,017
2 votes

Unsure about step in dervation of Arnoldi iteration (from Fundamentals of Numerical Computation)

Since $q_{1},\ldots,q_{m}$ span the same Krylov space as $b,Ab,\ldots,A^{m-1}b$, it follows that $Aq_m$ lies in the next Krylov space. We want $q_{m+1}$ to orthonormally extend the basis to that space ...
Toby Driscoll's user avatar
2 votes
Accepted

Projecting a symetric matrix on a subspace of known kernel

Without loss of generality you can assume that $u,v,w$ are orthonormal. If not, use Gram Schmidt. Then complete it with an orthonormal basis $f_4,\ldots,f_n.$ Write the orthogonal matrix $U$ which ...
Gérard Letac's user avatar
2 votes
Accepted

Recurrence relation from Thomas Algorithm

Letting $u=1+\frac1x,$ and $g_n=-\frac1{f_n},$ you get: $$g_0=u, g_{n+1}=2u-\frac{1}{g_n}.$$ We easily see that $g_n$ is the ratio of polynomials of $u,$ with the numerator of $g_n$ becoming the ...
Thomas Andrews's user avatar
2 votes

Finding the bounds of the missing values of a symmetric positive semidefinite matrix

This matrix will fail to be positive definite except in the case that $x = 1$. One way to see this is to consider the Schur complement $$ \pmatrix{1 & x\\x & 1} - \pmatrix{1\\1}(1)^{-1}\...
Ben Grossmann's user avatar
2 votes
Accepted

Found Eigenvector is close to the solution but not correct, what step am I missing?

You found the eigenvalue $\lambda = 9$. You want to find a corresponding eigenvector, that is, a vector $x$ such that $Ax = 9x$, or equivalently, $(A-9I)x = 0$. Writing this as a matrix equation, $$\...
suhasiny's user avatar
2 votes
Accepted

Matrices that are simultaneously Cauchy matrices and Toeplitz ones

The Wikipedia article Cauchy-matrix states In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an $m\times n$ matrix with elements $a_{ij}$ in the form $$ a_{ij}= \frac {1}{x_i-...
Somos's user avatar
  • 35.7k

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