# Tag Info

Accepted

### 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 ...
• 12.8k
1 vote
Accepted

### 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.
• 1,590
1 vote
Accepted

### 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 ...
Accepted

### 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 ...
• 6,077
Accepted

### 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.

### 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 $$(I + A)x = b.$$ ...
• 12.8k

• 51
Accepted

### Regularization of Interest Rate Inverse Problem failing

use np.eye(M) for the identity matrix. ...
• 3,407
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 ...
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} \\ &= ...