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### 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 ...
• 7,244
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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 ...
• 12.9k
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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 ...
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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.
• 1,600

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