# 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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### Eigenvalues of the sum of a matrix such that all its principal submatrices are stable, and a diagonal matrix with non-positive entries

I have a real square matrix $A$ (not necessarily symmetric) with all its principal submatrices (including $A$) having eigenvalues with a negative real part. On the other hand, I have a matrix $D$ ...
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### An inequality of eigenvalues of $A$ and $U^TAU$, where $A$ is symmetric, $U^TU = I$.

The question is as follows: $A$ is a $J\times J$ symmetric matrix, $U$ is a $J\times K$ matrix where $K\le J$. Suppose that $U^TU=I_K$. Please show that $\lambda_{j}(U^TAU)\le\lambda_{j}(A)$ I have ...
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### Eigenvalues of a triangular matrix from one base to another

@Gerry Myerson in the comments bellow offered a better formulation to my question ...
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### Eigenvalue that depends on eigenvector

Consider the problem $$(\mathbf{1}^TAx) x=Ax,\tag{*}$$ where $A$ is a square matrix and $\mathbf{1}$ is a column vector of ones. It does not need to be of exactly this form, the main point is that we ...
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### Two eigenvalues and an eigenvector walk into a bar…

Suppose I have the transformation $T(v) = Av = \lambda v$. If two of the eigenvalues are $\lambda_1$ and $\lambda_2$ where $\lambda_1=-\lambda_2$, is there a way to quickly find the eigenvector(s) for ...
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### Characteristic Equation for Traceless Matrices

The characteristic polynomial for traceless $2\times 2, 3\times 3, 4\times 4$ matrices $A$ are \begin{align} x^2+&\det A \\ x^3-\frac{1}{2}{\rm Tr}A^2 x - &\det A \\ x^4-\frac{1}{2}{\rm Tr}A^...
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### What does it mean for Katz Centralities to “diverge”?

In Mark Newman's Networks book, 2010 edition, page 173, he explains some mathematical details behind the Katz Centrality measure: In matrix terms, Eq. (7.8) can be written x = αAx + β1, (7.9) where 1 ...
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### Show the eigenvalues of a square matrix are between zero and one.

Let $A$ and $B$ be two positive definite matrices of orders $n$ and $d,$ respectively. Let $X$ be a matrix of type $(n,d)$ ($n$ rows and $d$ columns). Is it true that $A^{-1} X B^{-1} X^\intercal$ ...
### Why is $\phi$ diagonalizable if $\phi \circ \phi =id_V$?
V is a finite-dimensional $\mathbb{Q}$- vector space with $\phi: V \rightarrow V$ Why does it follow that $\phi$ is diagonalizable if $\phi \circ \phi = id_{V}$? My ideas so far: I do know that if i ...