2
votes
1answer
27 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
0
votes
0answers
8 views

Find the eigenvector for an operator on a linear span

Let $V$ be the linear span of the functions $1,cos(x),sin(x)$. Let the operator $T$ on $V$ be given by the rule $Ty(x)=y(x+ \pi/4)$. Find the eigenvalues and eigenvectors of T in V. I know how to ...
0
votes
1answer
24 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
0
votes
0answers
24 views

Why is QR algorithm using plane rotation followed by givens rotation better than just plane rotations?

To find eigenvectors from a tridiagonal matrix, it[ref:Numerical Recipes] says that QR algorithm using plane rotation followed by givens rotation(QR algorithm with implicit shifts) better than just ...
0
votes
0answers
21 views

What do they mean by corresponds to the same eigenvector x in this question?

I'm getting confused by the wording of this question. What do they mean by corresponds to the same eigenvector x? Question: Suppose $\lambda$ and $\ell$ correspond to the same eigenvector x? Show ...
0
votes
1answer
22 views

eigen value is a 'continuous function' of matrices

I have a doubt in linear algebra basically about polynomials. If a sequence of real matrices $A_n$ converges to a matrix $A$, does it imply that in $\mathbb{C}^n$, the spectrum vectors $\sigma_n$ ...
1
vote
0answers
29 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
0
votes
1answer
23 views

Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

I'm having trouble with this problem : Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while : $$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$ ...
0
votes
0answers
17 views

About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms)

my question is originated from a physical problem. I will try to present the problem as simple as possible, but I fear it will still be long since I'm bad at expressing myself briefly. It starts with ...
2
votes
1answer
27 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
2
votes
3answers
62 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
1
vote
2answers
34 views

Why must every vector in V belongs to one of the generalised eigenspaces of $T: V \to V?$

Why must every vector in V belongs to one of the generalised eigenspaces of $T: V \to V?$ Is there a simple proof for this? Can someone provide me with an intuition behind it? Note that V is an ...
1
vote
3answers
47 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
-1
votes
1answer
31 views

Linear Algebra - Prove trival solution eigenvalue

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$. Prove that $(2A+5I)x=0$ has only trival solution. I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that ...
6
votes
1answer
55 views

What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
1
vote
1answer
47 views

Eigenspace conceptual question

Is it true that a vector space V is a direct sum of all its eigenspace? What happens if T is not diagonalisable? Does this only apply to a vector space over an algebraically closed field? Similar to ...
1
vote
1answer
47 views

Eigenvalues of tridiagonal matrix

on page 13 of the paper here there is a proof in theorem 4 that all eigenvalues of this tridiagonal matrix, which has strictly positive entries down the subdiagonals, are simple. Unfortunately, I ...
2
votes
1answer
20 views

Why can we solve eigenvalue problems which are non-convex by Lagrange multiplier methods and get global minima?

while reading the paper "Some Modified Matrix Eigenvalue Problem" by Golub this doubt occurred to me. there he writes that we can minimize $x^TAx$ subject to $x^TBx=1, Cx=0$ As far as I understand ...
1
vote
0answers
13 views

Why is the number of Jordan blocks with eigenvalue $\lambda$ and size $\geq l =$ dim (Ker $f_\lambda$ $\cap$ Im $f^{l-1}_\lambda$)?

Why is the number of Jordan blocks with eigenvalue $\lambda$ and size $\geq l =$ dim (Ker $f_\lambda$ $\cap$ Im $f^{l-1}_\lambda$)? Note that $f_\lambda$ = $f - \lambda I$. This is stated in my ...
0
votes
2answers
24 views

How to prove inverse of a linear operator is diagonalizable using concept of eigenspaces?

Let T be an invertible linear operator on a finite dimensional vector space V. Given for any eigenvalue $\alpha$ of T, $\alpha$^(-1) is an eigenvalue of T^(-1). I first proved that the eigenspace of ...
0
votes
1answer
27 views

Eigenvalues of linear transformations

Determine the eigenvalues of each linear transformation. Give brief explanations. (Hint: you do not need to find a matrix representing the linear transformation.) (a) $\mathcal P : \Bbb ...
1
vote
4answers
49 views

Linear Algebra - Prove Isomorphism.

Let $T : \Bbb R^n \rightarrow \Bbb R^n$ Linear transformation. Prove that there is a real number $\alpha$ that the transformation $\alpha I-T$ is isomorphism. isomorphism is only if $\ker T={0}$ or ...
0
votes
0answers
30 views

Generalizing the idea of Eigenvectors

I was thinking about the idea of an eigenvector as an element of a set that is closed under a transformation. So if $A$ represents the transformation, and if $\vec{u}$ is an element for which there ...
0
votes
2answers
30 views

Show that the inverse of a strictly diagonally dominant matrix is monotone

I have been struggling with this problem for awhile. Given that $A$ is a strictly diagonally dominant matrix with positive diagonal entries and non-positive off-diagonal entries, show that $A$ is ...
0
votes
0answers
26 views

Linear Alegbra - Is this linear transformation isomorphism?

Let $\mathbb R^4 \rightarrow \mathbb R^4$ linear transformation. That : $$\dim\operatorname{Im}(T+I)=\dim\ker(3I-T)=2$$ Is $T-I$ isomorphism? The only thing I come up with is that 3 and -1 are ...
1
vote
2answers
88 views
+50

Give conditions on a,b,c, and d such that A has two, one, and no eigenvalues?

I am given that matrix $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ and I need to find conditions on a,b,c, and d such that A has Two distinct ...
0
votes
3answers
62 views
+50

Find the eigenvalues of the matrix and give the bases for each of the corresponding eigenspaces

I'm having issues with this problem. I have solved for the eigenvalues but am having trouble finding the bases for both eigenvalues. The pictures below contain my work for solving for the eigenvalues ...
0
votes
1answer
48 views

Geometric Interpretation of Eigenvectors

I just want to make sure I'm thinking about this correctly. I've been given a matrix A and I need to find the eigenvalues and eigenvectors geometrically. I have found the eigenvalues. It wasn't too ...
2
votes
1answer
49 views

Linear Algebra question relating to eigenvectors

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs $(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal. Let $C = e_1e_1'+....+e_me_m'$. ...
0
votes
1answer
26 views

When does a square matrix have an eigen-decomposition? When is a matrix defective? [duplicate]

Some square matrices, like $ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$, don't have a complete set of eigenvectors. By complete I mean that the eigenvectors span the entire ...
1
vote
2answers
97 views

Possible eigenvalues of a matrix $AB$

Let matrices $A$, $B\in{M_2}(\mathbb{R})$, such that $A^2=B^2=I$, where $I$ is identity matrix. Why can numbers $3+2\sqrt2$ and $3-2\sqrt2$ be eigenvalues for the Matrix $AB$? Can numbers $2,1/2$ ...
4
votes
2answers
117 views

The rank and eigenvalues of the operator $T(M) = AM - MA$ on the space of matrices

This problem is from Artin Algebra Second edition, 5.2.3. Let $A$ be a $n\times n$ complex matrix. (a) Consider the linear operator $T$ defined on the space $\mathbb{C}^{n\times n}$ of all complex ...
1
vote
2answers
22 views

How to make a block matrix positive semi-definite?

I have a matrix $A=\begin{bmatrix} \textbf{0}_{N\times N} & S\\ S^T & \textbf{0}_{M\times M} \end{bmatrix},$ where $S\in R^{N\times M}$. What $S$ would make $A$ a positive semi-definite ...
0
votes
0answers
13 views

Discrete Fourier vectors are the eigenvectors for any linear, constant coefficient, periodic, finite difference discretization on a uniform grid?

I came across the following statement: It can be shown that the DF vectors are always the complete set of eigenvectors of any linear, constant coefficient, periodic, finite difference discretization ...
6
votes
2answers
115 views

Show that there do not exist 3 $\times$ 3 matrices $A$ over $\mathbb{Q}$ such that $A^8 = I $and $A^4 \neq I.$.

Show that there do not exist 3 $\times$ 3 matrices $A$ over $\mathbb{Q}$ such that $A^8 = I $and $A^4 \neq I.$. I am aware that the minimal polynomial of $A$ divides $(x^8−1)=(x^4−1)(x^4+1)$.If the ...
3
votes
2answers
56 views

Find eigen values of $B = \begin{bmatrix} 0 & A^* \\ A &0 \end{bmatrix}$

$$B = \begin{bmatrix} 0 & A^* \\ A&0 \end{bmatrix}$$ I think that $\det(B) = \det(A) * \det(A^*)$ and probably eigen values just get squared. What is the right answer? EDIT: ...
1
vote
1answer
29 views

Is it possible to obtain right eigenvectors from left eigenvectors under certain conditions?

Suppose we solved the eigenvalue problem $VA=\Lambda V$ and the resulting matrix of left eigenvectors $V$ is invertible. Then, diagonalize $A=V^{-1}\Lambda V$, multiply both sides by $V^{-1}$ to get ...
2
votes
1answer
36 views

Average value of a bilinear map on a Euclidean sphere

Let $(V, g = \langle \cdot, \cdot \rangle)$ be a Euclidean vector space and $B : V \times V \to \mathbb{R}$ be a symmetric bilinear form. I would like to know if something like this is true: ...
2
votes
1answer
25 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
0
votes
0answers
3 views

Spectral value relation between CVD and DVC , V orthagonal and C,D diagonal

Consider an orthogonal matrix V and two diagonal matrices D and C as same oder as V. We know that eig(CVD) = eig ( DVC ). Is there any result relating the singular values of CVD and DVC for example ...
2
votes
1answer
59 views

Simplifying a characteristic equation when one eigenvalue is known

This is either trivial, or difficult; if the former I should be embarrassed to ask it. Anyway... I have a 4x4 matrix of non-zero integer values, for which the determinant is zero. Given that ...
0
votes
2answers
30 views

Prove a quadratic form is positive definite

I want to prove - without using eigenvalues- that the quadratic form $$q(x,y)=Ax^2+2Bxy+Cy^2$$ is positive definite iff $A>0$ and $AC-B^2>0$ This exercise was taken from a practice for a ...
0
votes
1answer
25 views

Eigenvectors for shear matrix and diagonalizing.

Here is a shear matrix $ \begin{pmatrix} 1 && 0 \\ 2 && 1 \end{pmatrix}$. The eigenvalues are 1. $ \lambda^2 - 2 \lambda + 1 \to \lambda = 1$. So now I try to find the eigenvectors. ...
1
vote
2answers
57 views

How to find eigenvectors of this matrix

I want to find eigenvectors of the following matrix manually. $$ A = \begin{bmatrix} 300 & 100 & 75 \\ 100 & 200 & 50 \\ 75 & 50 & 100 \end{bmatrix} $$ I found eigenvalues ...
7
votes
2answers
115 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
0
votes
1answer
21 views

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix. $V = P_1(R)$ and $T(ax+b) = (-6a + 2b) x + (-6a + b)$ First i gave the canonical basis of ...
0
votes
1answer
27 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...
0
votes
0answers
30 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
0
votes
0answers
27 views

If I have a real symmetric matrix and one of the diagonal elements are zero, does that say anything about the eigenvalues?

In the two dimensional case, if I have a matrix $\left[\begin{array}{cc} 0 & a\\ a & b \end{array}\right]$ or $\left[\begin{array}{cc} b & a\\ a & 0 \end{array}\right]$ then I have a ...
5
votes
2answers
46 views

Finding the eigenvalues of a given Markov matrix

Let $$A = \begin{pmatrix} 0.6 & 0.1 & 0.1\\ 0.1 & 0.8 & 0.2\\ 0.3 & 0.1 & 0.7 \end{pmatrix}$$ I want to find the eigenvalues of this matrix. Because this is a markov matrix, ...