# Questions tagged [eigenvalues-eigenvectors]

Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.

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### How to solve this Linear Algebra question? [closed]

Let \begin{pmatrix} 0 & 0 & 0 & a \\ -1 & 0 & 0 & b \\ 0 & -1 & 0 & c \\ 0 & 0 & -1 & d \end{pmatrix} Suppose 0(Zero) is an eigenvalue of A with ...
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### How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix

If D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
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### Eigenvalues of A, B, and (A+B) [closed]

Could the following statement be correct? If all the real parts of the eigenvalues of matrices A and B are negative, then all the eigenvalues of (A+B) also have negative real parts. Alternatively, the ...
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### polarization ellipse for complex eigenvalues corresponding the phase and eigenstates.

I want to draw polarization ellipses at 0.0 eV, 0.12 eV, 0.16 eV, and 0.2 eV for my transmission eigen-polarization-values plots using eigenphase data (in radians). I've attached the final result ...
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### Prove that a square rank $1$ matrix can have at most one eigenvalue different from $0$

I'm sure this can be proven in other ways, but I'm curious if the gap in the line of reasoning presented below can be filled so that the proof is valid. Proof by contradiction. Assume the opposite: ...
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### Eigenvectors of two commuting diagonalizable matrices when the eigenspaces need not have dimension one

Let $A,B$ be commuting diagonalizable $n\times n$ matrices over $\Bbb C$. Suppose that the eigenvalues of $A$'s are all distinct (so the eigenspaces have dimension one), and the same for $B$. Then any ...
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### If $\lambda$ is an eigenvalue of $T$, then $|\lambda|^{2}\le\displaystyle\sum_{j=1}^{n}\sum_{k=1}^{n}|M(T)_{j,k}|^2$

This is a problem from Axler's "Linear Algebra Done Right", 4th Edition, problem 19 of section 6A: Suppose $v_1,\dots,v_n$ is a basis of $V$ and $T\in L(V)$. Prove that if $\lambda$ is an ...
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### compute eigenvectors of an $n\times n$ linear transform knowing the eigenvectors of an $(n-1)\times(n-1)$ linear transform on a projected subspace

Supposed I'm trying to find the eigenvectors and eigenvalues of a square $n\times n$ matrix $A_{n}$. Further suppose that I've applied some algorithm and identified one eigenvalue and eigenvector ...
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### Eigenvalues of a circulant matrix

So we define a circulant matrix as follows. Let the first column be $$a^t = [a_0, a_1, a_2, a_3, \dots,a_{n-1}].$$ The other entries are given by circularly permuting the column 1. For example column ...
For which values of $a\in\mathbb{R}$ are the following matrices congruent? $$A=\begin{pmatrix} 1&4-a-a^2\\ 2& -1 \end{pmatrix}$$ $$B=\begin{pmatrix} -a-1 & 3\\ 3 & -5 \end{pmatrix}$$ ...