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2 votes
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Solve the matrix equation $A = - B A B^T + C$ without matrix inversion or vectorization

This is the Stein matrix equation also known as the discrete time Lyapunov matrix equation. Your analysis using the Kronecter product form shows that the equation has a unique solution $X$ for every ...
Carl Christian's user avatar
1 vote
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Kalman Filter -- handling large covariance matrices with principal-component-like structures

There are some "rank-reduced Kalman Filter" but they look much more complicated then the classical version It does look like ensemble kalman filter is more generic.
CuriousMind's user avatar
  • 1,590
1 vote
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How to find the eigenvector of symmetric tridiagonal matrix

Assuming $b\neq 0$, the eigenvectors are independent of $a$ and $b$ and given by $v_k=(v_{k1},…,v_{kn})$, where $v_{kj}=\sin(\pi kj/(n+1))$. Here, $n$ is the size of the matrix and $k=1\dots,n$. You ...
Hjalmar Rosengren's user avatar
2 votes
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Sensitivity of the eigenvalues to a change in a diagonal element.

Since you are discussing adjacency matrices, I will assume that all of these matrices are symmetric. Let $A$ be a $n\times n$ real and symmetric. It then has $n$ real eigenvalues which we order like ...
whpowell96's user avatar
  • 6,077
0 votes
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Matlab qz algorithm not reliable

The 5 significant figures fooled me. Both mine and the matlab qz algorithm reached the same conclusion, differing only by a huge constant. Therefore I revoke my criticism of the matlab qz function.
Littlejacob2603's user avatar
2 votes

If I can solve $Ax = b$ efficiently, is there a method to solve $(I+A)x=b$ efficiently?

It is doubtful in the extreme that the ability to solve linear systems of the form $$Ax=b$$ efficiently can generally be used to accelerate the solution of linear systems of the form $$(I + A)x = b.$$ ...
Carl Christian's user avatar
0 votes

Show backward Euler method is $0$-stable

This is an old question, but since I was working on the very same problem, I will add some comments here for future readers. I continue from where you left off (replacing $\vec{s}_n$ with $\mathbf{e}...
S4JJ4D's user avatar
  • 111
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
0 votes

Are all matrices equipped by definition with a matrix-vector product?

You will find that if you trace such ontological dependencies long enough, that what you get a net that has cycles and worse. To get a hierarchy of concepts one has to cut the net to get a spanning ...
Lutz Lehmann's user avatar
1 vote

How can I determine a linear recurrence relationship given a matrix raised to integer powers?

I think I've found an answer to my own question based on some of the comments made on the original question. $A$ is an $m \times m$ matrix. Using the Cayley-Hamilton theorem, I can express $A^m$ as: $...
ngolds's user avatar
  • 51
0 votes
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Regularization of Interest Rate Inverse Problem failing

use np.eye(M) for the identity matrix. ...
Yimin's user avatar
  • 3,407
0 votes

Is the matrix positive definite given the Gauss-Seidel method converges?

It's right. When $A$ is invertible and symmetric with positive diagonals, we have "Gauss-Seidel method converges iff $A>0$". Only prove the converse side. Let $A=D-L-L^T$ where $D$ is ...
BlowingWind's user avatar
1 vote

Binary matrix power for a specific entry.

Let's look at the formula for increasing $k$, \begin{align} (A^1)_{i, j} &= A_{i, j} \\ (A^2)_{i, j} &= \sum_m A_{i,m} A_{m,j} \\ (A^3)_{i, j} &= \sum_{m} (A^2)_{i, m} A_{m,j} \\ &= ...
Tony Mathew's user avatar
  • 2,407

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