# Questions tagged [numerical-linear-algebra]

Questions on the various algorithms used in linear algebra computations (matrix computations).

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### What is the "lifted system of linear equations"?

I am reading this blog post, where it says Here’s another variant of the same idea. Suppose we want to solve the linear system of equation $(D + uv^\top)x = b$ where $D$ is a diagonal matrix. Then we ...
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1 vote
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### Is this generalization of determinant for a higher-order tensor a standard object?

The determinant of an $n$ by $n$ matrix $a$ can be defined as $$\mathrm{det}(a)= \sum_{\sigma} \mathrm{sgn}(\sigma) a_{1,\sigma(1)} a_{2,\sigma(2)} \dots a_{n,\sigma(n)}$$ where $\sigma$ is a ...
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### Estimation of the eigenvalue of a matrix [closed]

Let $A_n$ be a $n×n$ matrix with entries $a_{ij}=n-|i-j|$, let $\lambda_n$ be the largest eigenvalue of $A_n$, find $\lim_{n \to \infty} \frac{\lambda_n }{n^2}$ I find it hard to compute the ...
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### Lanczos convergence for symmetric matrix with eigenvalues $1,2,\cdots,2,100$

Suppose that $A$ is symmetric with eigenvalues $1,2,\cdots,2,100 \in \mathbb{R}^{100x100}$ and $b \in \mathbb{R}^{100}$ obtained by normalizing a standard normal random vector. Show that 1 and 100 are ...
• 5,164
1 vote
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### Compressed image using SVD draws a clear line between part that's blank and part with a drawing. Why? [closed]

I'm trying to compress grayscale images using SVD. This is the original image: Yes, there's a lot of blank space. I then choose the x% largest singular values, perform the transformed matrices ...
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### Why everyone talks SOR and nobody JOR?

Context I am considering both Jacobi and Gauss-Seidel to be well-established. Essentially, given a relaxation parameter $\omega$, we have the following two methods (we will indicate the result of ...
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### QR algorithm, equivalence of r shifts and r times 1 shift

Is it true that in general the general $QR$ algorithm applied to $A$, a multiple shift of degree $r$ is equivalent to a sequence of $r$ single shifts of degree $1$? (Assuming that no shift is an exact ...
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1 vote
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### Matlab qz algorithm not reliable

I programmed my own version of the qz algorithm and, while testing it's results using the matlab qz algorithm, I found a particular case where my solution reaches an upper-triangular matrix and matlab ...
1 vote
133 views

### Sensitivity of the eigenvalues to a change in a diagonal element.

I'm currently examining irreducible diagonally dominant matrices say A, i.e. $\exists i$ $|a_{ii}| > \sum _{i \not = j} a_{ij}$ and the corresponding graph G to the adjacency matrix A is strongly ...
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### Kalman Filter -- handling large covariance matrices with principal-component-like structures

I understand that the estimation of the covariance matrices are the important part of the Kalman filter. However in my use case my covariance matrices are really big but with a pretty neat factor ...
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