# Questions tagged [eigenvalues]

Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).

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### An Upper Bound of $\left<H,X\right>_F$ with Constrainted Rank

We have $X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}$, $H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}$, and $d<n$. $H$ has rank $r\leq d$ and $X$...
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### Eigenvectors associated to two quasi-degenerate eigenvalues

I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$. Within Matlab enviroment, I use the command ...
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### Numerically find the nearest positive semi definite matrix to a symmetric matrix

I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
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### Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
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### How does the QR algorithm applied to a real matrix returns complex eigenvalues?

I'm a noob into eigenvalues algorithms, but something call my attention. QR algorithm works with real/complex matrices producing real/complex eigenvalues. However, it can not produce complex ...
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### Derive the formula for eigenvalues

If $A$ has eigenvalue $\lambda_A$ $$B = I - c\frac{I-rA}{I-\bar{r}A}$$ How to derive the eigenvalue $\lambda_B$? $$\lambda_B=1-c\frac{1-r\lambda_A}{1-\bar{r}\lambda_A}$$ where $c, r, \bar{r}$ are ...
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### LAPACK sorting eigenvalues differently each time

I'm using LAPACK zgeev routine to get eigenvalues and eigenvectors of a symmetric matrix in C++. Problem is zgeev is being called in a loop but it sorts eigenvalues (and eigenvectors) differently ...
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### Matlab, Mathematica & LAPACK returning 3 different eigenvectors

(I'm not sure which of math.se / stackoverflow / scicomp.se is the right place to ask this question) I have a C++ code which generates a complex matrix and then calculates its eigenvalues and ...