# 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 $\lambda x - Tx = 0$.

10,996 questions
Filter by
Sorted by
Tagged with
25 views

### Are all eigenvalues of $((C^TQ_fC)^{-1} (C^TQ_gC))$ lie in $(0,1]$?

Consider a matrix $X_f = (C^TQ_fC)^{-1}$ and $X_g = (C^TQ_gC)^{-1}$, where $C \in \mathbb{R}^{n \times m}$ is a full column tall matrix ($m < n$). $Q$ ($\in \mathbb{R}^{n \times n}$) is a diagonal ...
19 views

### Find Square root of a matrix from its spectral decomposition

Let $\rho$ be $n\times n$ symmetric matrix, thus the spectral decomposition of $\rho$ is $$\rho=\sum _{i=1}^n e_i\left|\psi _i\right\rangle \left\langle \psi _i \right|$$ where $e_i$ and $\psi _i$ ...
16 views

### Eigenvalues of the sum of a positive and a positive semidefinite matrix

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times n}$ respectively a positive and a positive semi-definite matrix. Is it possible to establish an upper bound for the minimum eigenvalue of the ...
13 views

### “Diagonalisation” of second order linear PDEs

Define the operator $$Pu=\sum_{ij} a_{ij}(x) \partial^2_{ij} u+\sum_k b_k(x) \partial_k u+c(x) u.$$ My question is, in what situation can we somehow "diagonalise" the principle part of $P$,...
33 views

### Is there a geometric interpretation of eigenvalues of integer matrices?

In some instances, like physics, you may find that quantities you are after are eigenvalues of matrices. However, for example, explaining that "the mass of a muon is an eigenvalue of a matrix&...
63 views

### Commuting Matrices with Complex Eigenvectors

Main Question Say that $K$ and $S$ are two commuting matrices with a full set of real eigenvectors, where the dimension of the eigenspaces of the matrices may be greater than one. $$KS=SK$$ It is a ...
46 views

### Show that $T \in \mathcal{L}(\mathbb{R}^4)$ with det $T<0$ has at least two distinct eigenvalues.

a) Suppose that $T \in \mathcal{L}(\mathbb{R}^4)$ satisfies det $T < 0$. Show that $T$ has at least two distinct eigenvalues. b) Find a $T \in \mathcal{L}(\mathbb{C}^4)$ where det $T < 0$ such ...
58 views

### Eigenvector of a matrix of all 1's

Consider the matrix $A \in \mathbb{R}^{n \times n}$ of all ones. Because there is only 1 linearly independent column, there are $n-1$ zero eigenvalues and 1 non-zero eigenvalue which is $n$. So one ...
20 views

### What is a simple definition of Lanczos iteration that is understandable?

I have seen that Lanczos's Algorithm can be used to tri-diagonalize a matrix but all of the definitions I have seen of it have been very complicated to understand.
20 views

23 views

### Relationship between eigenvectors of a covariance matrix and its component covariance matrices

I have a covariance matrix $\mathbf{S} = \mathbf{X}^T \mathbf{X}$. I then divide the matrix $\mathbf{X} \in \mathbb{R}^d$ that has $N$ rows into $n$ portions $\mathbf{x_i} \in \mathbb{R}^d$ each ...
48 views

### Question about a proof of the Perron Frobenius Theorem

I am going through a proof of the Perron-Frobenius theorem that I found online, but I'm having trouble understanding some of the steps. Here is the relevant info: Let $T$ be a primitive row-stochastic ...
23 views

### Eigenfunctions and eigenvalues of bidimensional operator in $Z^+$

I'm dealing with some discrete operators and I'm having some difficulties to find an expression or a method to find a set of eigenfunctions and eigenvalues for the bidimensional operator $S$ defined ...
40 views

### Are all strictly positive semi-definite matrices singular?

If I have some matrix A with an eigenvalue of 0, what makes this matrix singular? and I am assuming All positive definite matrices are non singular so all strictly positive-semidefinite matrices would ...
59 views

### Eigenvalues with positive real part for a matrix product

Let $M \in \mathbb{R}^{d\times d}$ be an invertible real matrix (not necessarily symmetric), and assume $M$ is positive semi-definite in the sense that $$v^T M v \geq 0$$ for all $v \in \mathbb{R}^d$...
40 views

### A question based on linear transformation from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ with its 2 eigenvalues given

This question was asked in my linear algebra quiz and i am unable to solve it . Let $A: \mathbb{R}^{2} \to \mathbb{R}^{2}$ be a linear transformation with eigenvalues $2/3$ and $9/5$ . Then , show ...