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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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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36 views

Are $A$ and $A^\top$ similar?

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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35 views

Largest eigenvalue of a Hermitian matrix

I have two positive semi-definite Hermitian matrices $\mathbf{R}_1, \mathbf{R}_2 \in \mathbb{C}^{M \times M}$. They are in fact covariance matrices. I'm interested in the largest eigenvalue (or ...
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1answer
18 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
46 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
44 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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35 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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21 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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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
35 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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70 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
26 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
34 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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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
34 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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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
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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 ...
3
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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
27 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
57 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
55 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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$A v = \lambda v \implies A^* v = \bar{\lambda} v$ 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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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
50 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 & ...
0
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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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398 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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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 ...
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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. ...
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1answer
41 views

Finding the Jordan Form and basis

$$A= \begin{pmatrix} 2&1&2\\ -1&0&2 \\ 0&0&1 \end{pmatrix}$$ I found that $$f_A(x)=m_A(x) = (x-1)^3.$$ So the Jordan form must be: $$J= \begin{pmatrix} 1&0&0\\ ...
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1answer
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matrix with eigenvalues on the unit circle

Suppose $A$ is an $n\times n$ matrix with complex entries such that there exists strictly positive constants $c_1<1<c_2$ so that $$c_1<\frac{\|A^Nx\|}{\|x\|}<c_2$$ for any integer $N\geq ...
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Determine if a basis consists of eigenvectors

So, this might be a silly question, but here it is. I am doing a couple of problems computing $[T]_\beta$, and determining whether $\beta$ is a basis consisting of eigenvectors of $T$. My problem is ...
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1answer
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The Adjacency Matrix of Symmetric Differences of any Subset of Faces has an Eigenvalue of $2$…?

Assume a planar graph $G$ and let's call its faces $f_k\in F$. The adjacency matrix of any face $f_k$ has an eigenvalue of $2$, since it's a $2$-regular graph, i.e. a cycle. I want to show that the ...
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Transformations with invertible matrix representations are onto

Let $L \colon \Bbb R^n \to \Bbb R^n$ be a linear mapping and let $B$ be a basis for $\Bbb R^n$. Prove that if $[L]_B$ is invertible, then $\operatorname{Range}(L) = \Bbb R^n$. I really can't see ...
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Eigenvectors of real symmetric matrices are orthogonal (more discussion)

This is an old question, and the proof is here The proof assumed different eigenvalues with different eigenvectors. My question is how about the repeated root? How to guarantee there will not have ...
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1answer
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why one of eigen values of a covariance matrix is zero?

let say you have a square matrix A, you calculate covariance of it, then calculate eigenvalues and eigen vectors. It follows that one of eigen values is equal to zero. why is it so? what does it mean ...
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Problem Involving Eigen Functions/Values in Differential Equation

I am confused about finding eigen values/functions for the following exercise. $$y'' - \lambda y = 0 , y(0) = 0, y'(L) = 0 $$ When $$ \lambda =0 $$ I find that $$ y = c_1cos(x) + c_2sin(x) $$ $$ y' ...
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Solving a BTTB system by BCCB extension that is highly structured and fewer degree of freedom

Consider a BTTB system generated by a simple $3\times 3$ matrix, $$ Col_1 = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ ...
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1answer
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IMC 2011 Day 1 Problem 2 - Linear Algebra

I have been reading the solutions of a past IMC paper (from 2011, Day 1) and I did not understand the solution to Problem 2 completely. Problem 2: Does there exist a real $3$x$3$ matrix $A$, such ...
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Computing the eigenvalues of $\mathbb{1}-I$

Let $A=\mathbb{1}-I \in \{0,1\}^{n \times n}$, the matrix having 0 in the diagonal and 1 everywhere else. To compute the eigenvalues I tried to compute the characteristic polynomial using recursion, ...
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Is it possible to get the original eigenvector after scaling a matrix?

Let ${\mathbf{X}}\in\mathbb{R}^{n\times 1}$ and ${\mathbf{Y}}\in\mathbb{R}^{n\times 1}$ and let $\mathbf{A}\in\mathbb{R}^{n\times 2}$ be defined as \begin{equation} \mathbf{A} = ...
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Spectral radius of a real, symmetric, positive semi - definite matrix.

While answering a question, the OP made a follow - up question, that I was not able to answer at that moment. However, I came up with an intriguing (at least to me) question. Let ...
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Eigenvalues of a Block Matrix from Schur Determinant Identity

I'm recently playing with some $(n+k)\times (n+k)$ block matrices: \begin{equation*} \mathbf{X} = \begin{pmatrix} \mathbf{0}_{n\times n} & \mathbf{P}_{n\times k} \\ {\mathbf{P}}_{k\times n}^T ...
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3answers
60 views

Find $P$ such that $P^{-1}AP = J$

Let $$A = \begin{bmatrix} 1 & 1 & 0 & -1 \\[0.3em] 0 & -1 & 1 & 2 \\[0.3em] -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
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1answer
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Example of non-normal operator whose all eigenvalues are real

Does there exist a non normal operator whose all eigenvalues are real.
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Eigenvalues of Hermitian matrix, reflexion matrix

If given me Hermitian matrix, with eigenvalues $1, -1$. Is it necessarily a reflection matrix? The matrix: $$\frac15\begin{pmatrix}-3 & 4i\\-4i & 3\end{pmatrix}$$ It is possible to say that ...
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Perturbed density of eigen-states of a 3 diagonal matrix

How does the density of eigen-states ($D(\lambda)$ is defined as $D(\lambda) d\lambda$ = Number of states in the range $\lambda ... \lambda + d\lambda$) of the following tridiagonal matrix ($A$) ...
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1answer
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Generalised eigenspace decomposition

I'm building up to the study of Jordan normal form, and need to prove that a vector space can be decomposed as a direct sum of generalised eigenspaces of a linear map. The proof in my notes goes: $$ ...
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Succinct notation for specifying that eigenvalues must have negative real part?

Is there a succinct way to denote that all eigenvalues of a matrix $A$ have negative real parts? If the eigenvalues were real, I could simply write this as $$-1 < A < +1$$ since we have the ...
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1answer
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The product of two positive definite matrices has real and positive eigenvalues?

Given two real positive definite (and therefore, symmetric) matrices $A$ and $B$, are all the eigenvalues of $AB$ real and positive? Wikipedia says $AB$ is positive definite if $A$ and $B$ are ...