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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Does every invertible complex matrix have an eigenvector? [on hold]

Over $\mathbb{C}$ does every invertible matrix have at least one non-zero eigenvalue and an eigenvector? I'm generally confused about eigenvectors and eigenvalues. I understand that eigenvectors are ...
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1answer
19 views

Eigenvalues of Certain Symmetric Block Matrix

What can we say about the relation between the eigenvalues of the following block matrix with identity diagonal blocks, and the singular values of the off-diagonal blocks: \begin{equation} ...
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1answer
17 views

Interlacing Theorem on Singular Values

Does the Cauchy's interlacing theorem hold for "singular values" of matrices too? I saw on this publication first Theorem that it does. It states that singular values of a matrix interlace the ...
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1answer
19 views

Special Properties of Real Matrices With Real Distinct Eigenvalues

Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do ...
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0answers
8 views

Condition number of positive definite matrix after rectangular orthogonal transformation on both sides

What is a lower bound on the condition number of $B A B^{T}$ (besides the trivial $\operatorname{cond}(B A B^{T}) \ge 1$) where $A$ is an $n \times n$ symmetric positive definite matrix, $B$ is a ...
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0answers
34 views

Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
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1answer
47 views

Is it possible to find a companion matrix of a polynomial which is also hermitian?

The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial $$ a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ I can ...
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2answers
37 views

Suppose that $u,v \in \mathbb R^n$ with $u,v$ not equal to $\mathbf 0$, and let $A= I + uv^\top$.

a) Show that $1+v^\top u$ is an eigenvalue of $A$ and $u$ its eigenvector. b) Define the subspace $S$ of $\mathbb R^{n}$ to be $$S=\{x \in \mathbb R^{n}\mid v^\top x=0\}= \operatorname ...
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1answer
40 views

Computing eigenvalue of the adjacency matrix of a path

Let $A\in \{0,1\}^{n \times n}$ be the adjacency matrix of a path of length $n$, i.e. having ones on the two off-diagonals, and zeros elsewhere. How does one compute the eigenvalues of this? I know ...
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1answer
24 views

Determinant of a Certain 3 by 3 Block Matrix with Scaled Identity Blocks

What is the determinant or/and eigenvalues of the following 3 by 3 block matrix: $$\left[\begin{array}{ccc} \frac{3}{4}I & \frac{1}{4}I & \frac{1}{4}I \\ \frac{1}{4}I & \frac{3}{4}I & ...
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0answers
21 views

Eigenvalue of Block matrix: Adjacency of complete bipartite Graph

Let $A\in \{0,1\}^{mn \times mn}$ be the adjacency matrix of a complete bipartite graph with $m$ and $n$ vertices each, i.e. let $A$ be the matrix consisting of two blocks $A_1\in \{0,1\}^{m \times ...
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1answer
20 views

Geometric and Algebraic Multiplicity, zero dimensions

The eigenvalues are $\lambda =0$(because we have multiplication here), $\lambda =1$, and $\lambda =2$ for the given characteristic equation, and as (a) states, that $GM\le AM$. Now, I want to know ...
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0answers
31 views

Prove $A,B$ share an eigenvalue [duplicate]

Let $A, B, C \in M_n(\mathbb{C})$ (not zero matrices) and let $g(x)\in\mathbb{C}[X]$. Let's assume $AC=CB$. Prove that $A,B$ share an eigenvalue. Things I've already proved (followed by the ...
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4answers
77 views

Prove that $g(A)$ is an invertible matrix

Let $A\in M_n(\mathbb{C})$ and let $\lambda\in\mathbb{C}$. Prove that if $\lambda$ is not an eigenvalue of $A$ then $A-\lambda I$ is invertible. Moreover, for $g(x)\in \mathbb{C}[x]$, prove that if ...
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0answers
19 views

Show that every Jordan matrix has a cyclic vector

Is my following reasoning correct? Since an $n\times n$ Jordan matrix has rank $n-1$ (because we can only make the last row the zero row), its geometric multiplicity is 1, which means the matrix has ...
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0answers
38 views

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
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1answer
61 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
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18 views

How to compute the Eigenvectors for a Markov matrix?

I have the following matrix for which I want to get the Eigenvectors. I know how to compute the Eigenvalues, but when I compute the vectors in the null space of the matrix, I get the wrong answer. ...
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0answers
54 views

Eigenvalues of $A=\begin{bmatrix}2&1\\\alpha&0\end{bmatrix}$

Determine (with respect to $\alpha $) all Eigenvalues (in $\space \Bbb C$) of the matrix: $$A=\begin{bmatrix}2&1\\\alpha&0\end{bmatrix}$$ We have: $$ \det(A-\lambda ...
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1answer
16 views

Sylvester's law of inertia for generic matrices.

By Sylvester's law of inertia, the positive and negative indices of a symmetric matrix $A$ are also the number of positive and negative eigenvalues of $A$. I was wondering if a similar result is known ...
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2answers
30 views

In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?

Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ...
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2answers
40 views

Sign of eigenvalues of $A$ by $\det(A-\lambda I)=\lambda \det(B+D-\lambda I).$

Let $A$ be a $n\times n$ matrix, $B$ be a $(n-1)\times (n-1)$ matrix and $D$ be a $(n-1)\times (n-1)$ diagonal matrix with all entries positive. We assume that $$\det(A-\lambda I)=\lambda ...
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1answer
46 views

Are $A$ and $A^\top$ similar? [duplicate]

Let $K$ be a field and $A$ a square matrix with entries in $K$. Then A and $A^\top$ have the same characteristic polynomial. What do we know about similarity? Do you have an example where $A$ and ...
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1answer
85 views

Largest eigenvalue of a Hermitian matrix

I have two Toeplitz positive semi-definite Hermitian matrices $\mathbf{R}_1, \mathbf{R}_2 \in \mathbb{C}^{M \times M}$. They are in fact covariance matrices satisfing the following conditions: (1) ...
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1answer
20 views

Eigenvalue replacing matrices in matrix equation?

Found this confusing pieces of equations in Liquid Crystal, from Fundamentals of Liquid Crystal Devices(2006), Wiley, Deng-ke Yang, Shin-Tson Wu. if A is an 2X2 matrix, from Cayley-Hamilton Theory A ...
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2answers
66 views

Finding Eigenvalue det(λI - A);

I want to know if what I'm doing to derive equation (2) from (M2) is correct or not; usually, before moving onto the next row in Guass-Jordan elimination we turn a_11 into a leading one or whatever ...
2
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2answers
50 views

Do linear operators that map one space into a different space have a Jordan canonical form?

I know that this answer is most likely "yes", and that, in the setting of matrices, all matrices are similar to its Jordan form, which is unique (up to the ordering of the Jordan blocks.) But what ...
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2answers
36 views

Eigenvalues of Matrix Product.

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their product? What about the special case when one of these matrices is a diagonal (positive) matrix? I ...
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2answers
22 views

Eigenvalues of a companion matrix

I've been tasked with the following: Show that the companion matrix $C(p)$ of $p(x) = x^2 + ax + b$ has characteristic polynomial $\lambda^2 + a\lambda + b$. Show that if $\lambda$ is an ...
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1answer
46 views

$6$ eigenvalues of a $4\times4$-matrix?

I am struggling with determining the eigenvalues of the following (symmetric) matrix: $$ A =\begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 ...
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1answer
38 views

Can the eigenvalues of a matrix always be expressed in terms of the traces of its powers?

In my research, I came across a cute identity involving the eigenvalues of a $2 \times 2$ Hermitian matrix $M$. These two eigenvalues can be expressed as follows: $$ \lambda = \frac{1}{2} \left[ ...
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2answers
74 views

Eigenvalues of the sum of two matrices: one diagonal and the other not.

I'm starting by a simple remark: if $A$ is a $n\times n$ matrix and $\{\lambda_1,\ldots,\lambda_k\}$ are its eigenvalues, then the eigenvalues of matrix $I+A$ (where $I$ is the identity matrix) are ...
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Is there a matrix with real entries such that $A \ne I_2$ but $A^3 = I_2$.

Is there a matrix with real entries such that $A \ne I_2$ but $A^3 = I_2$. I've actually encountered with this post: $A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$ ...
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2answers
27 views

equality of $\dim V_{\lambda_i}$.

Let a square matrix: $$Q = \left(\begin{array}{cccc} A&B\\0&C\end{array}\right)$$ $A,C$ are two square matrices such that $\lambda$ an eigenvalue of $A$ implies $\lambda$ isn't an eigenvalue ...
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1answer
40 views

Find a Basis $B$ of $R^2$ so that $B$ matrix of $T$ is diagonal

$T([1,1]^t) = [3,7]^t$ $T([1,-1]^t) = [1,1]^t$ Here's what I get: $T= \left(\begin{array}{cc}3 & 1 \\7 & 1\end{array}\right) $ The eigenvectors of $T$ is $E = \left(\begin{array}{cc} .4798 ...
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463 views

Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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1answer
38 views

Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$

I am wondering is it next true: Suppose that $f(t)$ is non-negative and non-decreasing function on $[0,\infty)$ and let $A$ be a positive operator on some infinite-dimensional separable Hilbert ...
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0answers
24 views

Describe the Jordan Normal Form of this operator,

In a previous question on MSE, I computed a 15x15 matrix of an operator. We see that the operator is nilpotent, with spectrum = {0}. But the last part of the problem asks to describe the Jordan ...
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2answers
88 views

How can I tell that my matrix is nilpotent?

I just computed a 15x15 matrix by hand :( It is not upper triangular as I hoped it would be. But my computations agree with what's offered in the student solution. My question is: the solution ...
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2answers
87 views

Prove that $V = \ker T \oplus \text{Im}T$

Let $T:V\to V$ such that $f_T = \sum_{i=0}^n c_ix^i$ and $c_1 = c_n = 1, c_0 = 0$. Prove that $V = \ker T \oplus \text{Im}T$. My thoughts so far: For some basis $B$, we have $[T]_B = A$. We know ...
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1answer
28 views

Shortcuts for computing the eigenvalues of a linear transformation

How would you calculate the eigenvalues of the following matrix? $A = \begin{pmatrix} -3 & 1 & -1 \\ -7 & 5 & -1\\ -6 & 6 & -2\end{pmatrix}$ $ $ $\ \ \ \ \ $$\chi_A(\lambda) ...
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2answers
58 views

Eigenvalue Problem — prove eigenvalue for $A^2 + I$

This is a proof I've been trying to figure out since the problem was presented to me. We are given that $\lambda$ is an eigenvalue for a matrix $A$ and the vector $u$ is the eigenvector corresponding ...
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1answer
56 views

Prove eigenvalue for $A^2 + I$ [duplicate]

This is a proof I've been trying to figure out since the problem was presented to me. We are given that $\lambda$ is an eigenvalue for a matrix $A$ and the vector $u$ is the eigenvector corresponding ...
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2answers
49 views

$A v = \lambda v \implies A^* v = \bar{\lambda} v$ if $A$ is normal [duplicate]

I want to show that if $A$ is normal then $$ A v = \lambda v \implies A^* v = \bar{\lambda} v $$ I can show that $A^*v$ is also an eigenvector of $A$, using the fact that $A$ and $A^*$ commute, but ...
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0answers
7 views

Is there an efficient $O(n^2)$ way to get the eigen decomposition given a LDL factorization?

Let's say I have a LDL factorization of a matrix A. Is there an efficient $O(n^2)$ way to get the eigen decomposition of A given it's LDL factorization? Is there a more efficient way, in case L and ...
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4answers
52 views

I'm trying to find the eigenvalues of a matrix. What is my mistake?

I have the matrix: $\left[ \begin{array}{cc} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{array} \right]$ which I rewrote as $\left[ \begin{array}{cc} λ-3 & ...
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2answers
38 views

Eigenvectors and eigenvalues of matrices

Say that we have a square matrix $M$, and that a non-zero vector $v$ can be an eigenvector of $M$ if $Mv = kv$ for some real number $k$. This real number, $k$, can be called the eigenvalue of $v$ with ...
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5answers
415 views

What is the intent of this problem, disguised as an eigenvalue - eigenvector problem?

Let $$ A= \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{bmatrix} $$ $a,b,c >0$. Find eigenvalues and a basis of ...
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0answers
22 views

Finding a jordan basis for a jordan form

Let $$A = \left(\begin{array}{cccc} 1&0&0&0\\3&-2&0&0\\14&0&-2&0\\8&-1&1&-2 \end{array}\right)$$ Easy to verify that $f_(x) = (x-1)(x+2)^3$. So the ...
3
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1answer
32 views

Eigenvalues of a binary matrix

Let $A = (a_{ij})$ be an $n \times n$ matrix with all entries equal to 0 or 1. Suppose that $a_{ii} = 1$ for $i = 1, \cdots, n$ and that $\det A = 1$. Then all the eigenvalues of $A$ are equal to 1. ...