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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Linear algebra: Complex eigenvectors

If $$A= \begin{pmatrix}1 & -1 \\ h^2 & 1\end{pmatrix},$$ I know the complex eigenvalues are $1+ih$ and $1-ih$. How do we find the complex eigenvectors? Can someone please explicitly show me ...
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20 views

Find the matrix $P$

$A= \begin{bmatrix}1 & -2 & 3\\-2 & 6 &-9 \\3 & -9 & 4 \end{bmatrix}$ Find $P$ with non-negative integer entries and has determinant $2$. $P^TAP=\begin{bmatrix}a & 0 ...
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1answer
15 views

Effect of spectral shift on the eigenvalues of a real symmetric matrix [duplicate]

Suppose a matrix A(real symmetric) is changed to A − σ I, where σ is any scalar quantity and I is the identity matrix. Explain what happens to the eigenvalues and eigenvectors of A? I am unable to ...
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42 views

Determinant and eigenvalues of Gram matrix lower bounds [on hold]

I'm trying to find a non-zero lower bound on the determinant of the Gram matrix $\Gamma$ assigned to linearly independent set of vectors (is there such a lower bound?). But that is not my question ...
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0answers
17 views

Finding the Eigenvalue of a general transformation

I'm studying for my Linear Algebra final and I'm having some issues with this proof: Given $T:V\rightarrow V$ $T=T^2$ what are all the possible Eigenvalues for T I'd appreciate any help you care ...
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2answers
32 views

eigenvalues of A - aI in terms of eigenvalues of A

I am stuck with this question of my assignment where given that A is nxn square matrix and a be a scalar it is asked to - Find the eigenvalues of A - aI in terms of eigenvalues of A. A and A - aI ...
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13 views

how to convert eigenvectors & eigenvalue to rotation matrix?

I would like to know how to convert an eigenvector and an eigenvalue(if needed) to a rotation matrix. I am in charge with writing software to calculate the attitude of a satellite in space. K is a 4 ...
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2answers
48 views

What are the possible eigenvalues of matrix $A$ that satisfies $A^2=-I$? [on hold]

Let $A$ be a matrix such that $A^2=-I$, where $I$ is identity matrix. What are the possible eigenvalues of $A$?
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27 views

When an eigen vector is zero vector

Question : Mike opens a bank account with an initial balance of 2000 dollars. Let b(t) be the balance in the account at time t. Thus b(0)=2000. The bank is paying interest at a continuous rate of 3% ...
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3answers
139 views

Minimal polynomial for an invertible matrix and its determinant

So here's one that I can't quite crack: Let $A\in M_n(\mathbb{F})$ be an invertible matrix with integer eigenvalues. Its minimal polynomial is ...
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1answer
16 views

No minimal polynomial for differentiation operator

Let $D$ be the Differentation operator of the of polynomials over $R.$ Prove that there is no polynomial $g(t),$ such that $g(D)=T_0.$ But characteristic polynomials satisfies it's operator. I dnt ...
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3answers
57 views

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$.

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$. here is the source of the problem. In the last ...
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3answers
34 views

Why is this matrix invertible [duplicate]

I was wondering if there is a way to see why $(1+A)$ invertible, if $A$ is a skew symmetric matrix. and I know that all eigenvalues of $A$ have zero real part and $A$ is unitarily diagonalisable.
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3answers
62 views

What are the possible eigenvalues of a linear transformation $T$ satifying $T = T^2$ [duplicate]

Let $T$ be a linear transformation $T$ such that $T\colon V \to V$. Also, let $T = T^2$. What are the possible eigenvalues of $T$? I am not sure if the answer is only $1$, or $0$ and $1$. It holds ...
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0answers
26 views

When do a Regular graph has an odd eigenvalue?

Merely looking at adjacency matrix of a regular graph, without explicit calculation, can we decide that graph will have an odd eigenvalue or not? If regularity is odd, we are sure that it will be an ...
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1answer
19 views

Find Matrix $S$ such that $S'CS=\left(\begin{array}{cc} 1_{N-1} & 0 \\ 0 & 0 \end{array} \right)$ where $C:=1_N-\iota \iota'$

The Centering Matrix $C:=1_N-\iota \iota'$ has eigenvalue $1$ of multiplicity $n − 1$ and eigenvalue $0$ of multiplicity $1$. Therefore a matrix $S$ with columns consisting of eigenvectors of $C$ can ...
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0answers
23 views

Eigenvectors of a real positive semi-definite submatrix

Let $\mathbf{A}$ be a real positive semi-definite matrix, and $\mathbf{V}$ and $\mathbf{\lambda}$ its eigenvectors and eigenvalues, respectively. I am wondering what is the relationship between ...
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22 views

Inital Value Problem from general solution

We have the following matrix: $\frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ And the inital condition: $\mathbf{Y_0} = (4,0)$ I have got the correct ...
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0answers
18 views

How to force a system of linear equations to return a non-trivial solution instead of the trivial one when finding the eigenvectors

When you've found the eigenvalues for a matrix A. And insert each eigenvalue into the relation: $(A-\lambda I) = 0$ and row reducing to echelon form for each one. In the situations where you end up ...
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2answers
44 views

characteristic polynomial and eigenvalues [on hold]

Find the characteristic polynomial and eigenvalues of $A=\left[\begin{matrix} 3 & 1 & 1 \\ 0 & 5 & 0 \\ -2 & 0 & 7\end{matrix}\right]$ $$ \begin{align*} \det(A-\lambda I) ...
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1answer
48 views

Easy Problem:Find eigenvalues of a $3 \times 3$matrix

Let $ A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 1 & w^2& w \\ 1 & w & w^2 \end{array} } \right] $ where $w$(other than 1) is a cube root of unity.Let ...
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2answers
27 views

Find the eigenvectors corresponding to an eigenvalue

I know how to find the eigenvectors corresponding to an eigenvalue of a matrix $A$: we basically need to find the vectors of the nullspace of $\lambda I - A$, but in my case, I have a matrix $A$ like ...
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2answers
29 views

When do two linear hermitian operators have a common eigen vector?

Let $H$ be an finite dimension hilbert space. Let $L_1$ and $L_2$ be two hermitian linear operators acting on this space. I know if these two operators commute they can be diagonalized in a common ...
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43 views

Find the eigenvalues for a matrix which is a product of matrices

Suppose I have a matrix $A \in \mathbb{R}^{2, 2}$ which is the product of $3$ other matrices, lets call them $A_1 = \left(\begin{matrix} cosx & -sinx \\ sinx & cos x\end{matrix}\right)$, $A_2= ...
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1answer
42 views

Zero characteristic polynomial?

Is it possible that characteristic polynomial of an $n \times n$ matrix be the zero polynomial? If this happens, this means that any scalar would serve as an eigenvalue?
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1answer
38 views

Problem on eigenvalues

Let A be real square matrix of order $n \geq 2$. Then show that: A. if $A^3 - I$ is singular, then $1$ is eigenvalue of $A$ B. if $A$ is singular, then $I+2A+A^2$ has eigenvalue $1$ My ...
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31 views

Eigenvalues & eigenvectors of a specific product of three matrices

How is possible, without multiplying, find the eigenvalues and eigenvectors of the A matrix ? Which propriety should I use? $$A= \begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & ...
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3answers
46 views

How to find eigenvalues $\lambda>0$ so that matrix A is positive and definite

We are given matrix A: \begin{pmatrix} s & -1 & -1\\ -1 & s & -1\\ -1&-1&s\\ \end{pmatrix} I need to find for which s do A has all eigenvalue $\lambda>0$(positive ...
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3answers
27 views

Eigenvalue eigenvector (basis)

I have a question regarding the use of eigenvectors as basis vectors. For A = \begin{bmatrix}1 & -4 &7\\-4 & 4 & -4\\7& -4 & 1 \end{bmatrix} "By expressing an arbitrary ...
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2answers
36 views

Determinant of symmetric matrix $(A-\lambda I)$

If we have a matrix $(A-\lambda I)$ which is: $\left( \begin{array}{ccc} 1-\lambda & -1 & 2 \\ -1 & 1-\lambda & 2 \\ 2 & 2 & 2-\lambda \\ \end{array} \right) $ Then it's ...
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The pair $x_1$ , $x_2$ are Linearly Independent

Prove that if $x_1$ and $x_2$ are eigenvectors with different eigenvalues, then the pair $x_1$ and $x_2$ are linearly independent. The way I went about this proof is by the contrapositive. So I am ...
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2answers
116 views

Linear Algebra - Eigenvectors and Eigenvalues

We are given the following problem: Consider the matrix $$ A = \left[\begin{array}{rrr} \cos\theta & \sin\theta\\ \sin\theta & -\cos\theta \end{array}\right] $$, where $\theta \in ...
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1answer
30 views

Linear Algebra - Eigenvalues and Eigenvectors

We are given the following problem: Suppose that $\theta\in\Bbb R$ is not an integer multiple of π. Show that the matrix $$ A=\left[\begin{array}{rrr} \cos\theta & -\sin\theta\\ ...
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4answers
41 views

Why if the columns of a matrix are not linearly independent the matrix is not invertible?

Why if the columns of a matrix are not linearly independent the matrix is not invertible? I have watched this video about eigenvalues and eigenvectors by Sal from Khan Academy, where he says that for ...
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2answers
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find the smallest interval in which the eigen value of the matrix lie

$$ \begin{bmatrix} 3 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 3 \\ \end{bmatrix} $$ I was practicing questions on Matrices & Determinants ...
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1answer
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Eigenvalues of an 2x2 matrix [closed]

How do i calculate the EigenValues of an Hessian Matrix which is 2x2.? And what is EigenValues. It is used in imagesProc. when i have to find goodFeatures is found when both eigenvalues are high.. but ...
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Signature Defect of a Matrix

Let $A,D$ be symmetric real matrices and let $B,C$ be real matrices such that $B$ and $C$ have the same number of rows, $A$ has the same number of columns as $B$, and $C$ has the same number of ...
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In which cases is the summation function distributive?

When working on the proof for $$\det(\text{A} -\lambda \text{I})=\det(\text{Q}^{-1}\text{ B Q}-\lambda \text{I})$$ where $\lambda$ is a scalar, $\det$ is the determinant, $\text{I}$ is the identity ...
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2answers
21 views

proof a theorem in linear algebra

prove that if λ1 and λ2 are two distinct eigenvalues of a matrix A and λ1 , λ2 are corresponding eigenvectors, respectively, then α1 and α2 are linearly independent please help... thank you...
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1answer
46 views

Eigen vectors for matrix with unknown constants?

I have the following matrix: $$\begin{bmatrix}\alpha&0&0\\\beta-\alpha&\beta&0\\1-\beta&1-\beta&1\end{bmatrix}$$ So far I have worked out the polynomial to be: ...
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0answers
15 views

Constrained zero diagonal low rank approximation of a matrix with zero diagonal

Suppose that you have a $n\times n$ matrix $A$ that is symmetric and has zero diagonal, such as for example $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0 & 1\\ 2 & 1 & 0}, $$ and you want ...
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1answer
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Question about eigenvalue of Hermitian matrix

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
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A Rayleigh quotient-related eigenvalue problem

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
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Find eigenvalues of a matrix using Perron–Frobenius theorem

I have to find the largest eigenvalue of a matrix containing only positive entries: $$\left( \begin{array}{ccc} e^{a} & 1 & e^{-a} \\ 1 & 1 & 1 \\ e^{-a} & 1 & e^{a} ...
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1answer
41 views

Gradient and Hessian of Abs(Non-Repeated Eigenvalue) of Non-Symmetric Matrix

I would like to compute in MATLAB, without resort to automatic differentiation), the gradient, and ideally also the Hessian, of the absolute value of a non-repeated eigenvalue of a non-symmetric ...
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38 views

Where am I going wrong with finding eigenvectors?

Simple example, but I am having the same issues with all of the problems I attempt. $$A= \left[\begin{array}{rrr|r} 6 & 3 \\ 2 & 7\\ \end{array}\right] $$ I get eigenvalues 4 ...
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28 views

linear algebra bound

I have a problem in course project? I have two positive definite $n\times n$ matrices, $A$ and $B$. I want to find the bound of singular values of a product of these matrices ? these matrices are ...
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1answer
34 views

Eigenvector of a $C^n$ class matrix

Let $A$ be the following matrix function: $\Bbb{R} \to \Bbb{R}^{a \times (a+1)}$ $t \mapsto A(t)$ Let us suppose that $A$ is $C^{\infty}$, meaning that all of $A$ coefficients are $C^{\infty}$. Let ...
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1answer
20 views

$A$ is singular and normal matrix, what must be its characteristic polynomial?

Let $A$ be a $5\times5$ real singular matrix which is normal. If $1-2i$ is an eigenvalue of $A$ and $2+i$ is an eigenvalue of $A^*$ (conjugate transpose), what must be its characteristic polynomial? ...
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33 views

How do we find eigenvalues from given eigenvectors of a given matrix?

For instance let $$A=\begin{pmatrix} 3 & -1 & -1 \\ 2 & 1 &-2 \\ 0 & -1 & 2 \\ \end{pmatrix}$$ be a matrix and $$u_1=\begin{pmatrix} ...