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4 votes
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What are the eigenvalues of a symmetric pentadiagonal Toeplitz matrix with zero tridiagonals?

Your matrix is equal to $$ -\,\,\begin{pmatrix} 0 &1 &&&&& \\ \newline -1 &0&1&&&&\\ \newline &-1 &0&1&&&\\ \newline &&...
Exodd's user avatar
  • 11.2k
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

How to find an exact solution for $X=X^T \in \mathbb{R}^{n \times n}$ satisfying $AX=XA^T$ and $B=XC^T$

In order to obtain a well conditioned equation I will assume that the state space model is minimal, so $(A,B)$ is controllable and $(A,C)$ is observable. Given is (minimal) state space model $$ \left\{...
Kwin van der Veen'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
  • 6,200
2 votes
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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
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
2 votes
Accepted

If $A$ is symmetric positive definite, then solving $Ax = b$ is equivalent to minimizing the quadratic form $\varphi(x) = \frac{1}{2}x^T A x - x^T b$

The proof is saying that when $Ax = b$, $\hat{\alpha} = 0$ is the value of the parameter of $\alpha$ that minimizes the expression $\varphi(x+\alpha p)$ and the value of this expression at $\alpha = \...
Wyatt Kuehster's user avatar
2 votes
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Solve ill-conditioned linear systems

Let $x$ denote the true solution of $Ax=b$ and let $\hat{x}$ denote the computed solution returned by our computer. By definition, the corresponding error $e$ is given by $$e = x - \hat{x}$$ and the ...
Carl Christian's user avatar
2 votes

Iteratively finding matrix inverse from a given inverted matrix.

If $A$ is a good approximation of $B$, then $A^{-1}$ is not necessarily a good approximation of $B^{-1}$. As an example consider the case of $$A = \begin{bmatrix} 1 & 0 \\ 0 & 10^{-9} \end{...
Carl Christian's user avatar
1 vote

How to find an exact solution for $X=X^T \in \mathbb{R}^{n \times n}$ satisfying $AX=XA^T$ and $B=XC^T$

Just drop for now the constraint that $X$ should be symmetric and consider the equation $AX-XA^T=0$. This is a Sylvester equation and the solution is unique if and only if $A$ and $A^*$ do not share ...
KBS's user avatar
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1 vote
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Using the definition, describe the backward error analysis and error bound of the eigen value problem.

No. Your use of the implication arrow is incorrect. Let $f : \mathbb{C}^{m \times m} \rightarrow \mathbb{C}^m$ denote a function such that $\lambda = f(A) \in \mathbb{C}^m$ is a vector that consists ...
Carl Christian's user avatar
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
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
  • 1,600

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