Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Kernel and image of this linear transformations

I dont know how i can solve this problem, if you know, please, explain me... help...: Consider in $\mathbb{R}^2$ the following linear transformations, $f_{\alpha}:\mathbb{R}^2 ...
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1answer
37 views

2014 IMC first problem first day (eigenvalues of a product of symmetric matrices).

This was the first problem of the IMC 2014. Let $A$ and $B$ be two $n\times n$ symmetric matrices with real entries which have all their eigenvalues strictly larger than $1$. Prove all the ...
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1answer
62 views

$A^tA-AA^t$ in Mathematical Physics

In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$ ...
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23 views

Efficient Test For Commuting Matrices

I know that if $A$ and $B$ are two Hermitian matrices, then $A B= B A$ if and only if their eigenspaces coincide [1]. In order to apply this test one need to compute eigenvectors of both $A$ and $B$ ...
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2answers
39 views

Can the difference between a number and a set of numbers be calculated as follows?

I have a number $X$ (lets say $45$) and I have five other numbers, lets say $(3, 4, ,5, 6, 7)$ I want to know the difference between my number and the other numbers like this: ...
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2answers
42 views

If then as matrix calculation

Having simple script code a like to bring a if-then-condition into linear algebra form. How is it made? Example 1: Having $T=25$ (where T is temperature current in room). If $T>30$ the equ. ...
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1answer
12 views

Bound on maximum of product of matrix and vector

I need to bound the absolute maximum of each entry of a matrix-vector product: $\max_{|x|_{1}=1} |Ax|_{\infty}$ I tried to pose this in terms of the induced infinity norm of $A$, as in ...
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1answer
19 views

Find basis corresponding to dual basis

In the finite dimensional vector space $V$, suppose $\{f_1,f_2,\cdots,f_m\}$ are the dual basis, how can find the basis $\{e_1,e_2,\cdots,e_m\}$ s.t. $f_i(e_j)=\delta_{ij}$
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3answers
268 views

How to find the limit of this matrix function

Let $A$ be $n\times n$ real symmetric matrix that is positive definite. Let $x\in\mathbb{R^n}, \space x\ne 0$. Prove that the following limit $$ \lim_{m\to\infty}\dfrac{x^TA^{m+1}x}{x^TA^{m}x} $$ ...
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0answers
33 views

What are the main kinds of mathematics? [on hold]

I stumble upon as much on math I don't know (trascendal math, number theory) and math I know on the internet and elsewhere. I have a pretty good idea about differential and integral calculus, and I'd ...
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1answer
26 views

How to solve an equation involving euclidean norm operation?

On page 3 of Scalable, Versatile and Simple Constrained Graph Layout it describes the equation: $$|(\mathbf p-\mathbf r)-(\mathbf q+\mathbf r)|=d$$ Where $\mathbf p$ and $\mathbf q$ are known ...
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28 views

Semplify $\det\left(D+M+A\right)$.

Let $D$, $M$, $A$, $n\times n$ matrices, with $n\in\mathbb N$. $D$ is a diagonal matrix, $M$ with elements all equal to $k\in\mathbb R$, $A$ is an antisymmetric matrix. Is possible to calculate ...
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3answers
157 views

Symmetric matrices and eigenvalues

If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$ I am trying to prove this as follows: If $v$ is an eigenvector of $A$, then $Av ...
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0answers
18 views

Find the population

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
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3answers
33 views

What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
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1answer
20 views

What can I assume, when given a matrix with information about its eigenvalues but not its action?

Basically, I've had to use linearity a couple of times yesterday and today, in order to write up a few proofs. But I notice that I am only given information such as positivity conditions and ...
2
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1answer
35 views

T-invariant subspaces

I'm studying for a qualifying exam, and I'd really appreciate some help on the following questions. Let p be a prime integer, $F=\mathbb{Z/pZ}$, V a vector space over F, and $T\in\mathscr{L}(V)$. ...
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25 views

Eigenspaces of $Y$.

Given an even dimensional real vector space and a complex structure $Y$, why is its complexification the direct sum of $\text{ker}(X + iY)$ and $\text{ker}(X - iY)$?
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3answers
93 views

If det $A = 0$ and $\det B \neq 0$ then show that $abc = -1$

This has been hurting my head for a while now.... If $$ \det\begin{bmatrix}a&a^2&1+a^3\\b&b^2&1+b^3\\c&c^2&1+c^3\end{bmatrix}=0 $$ And $$ ...
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0answers
20 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
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21 views

A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...
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0answers
20 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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2answers
48 views

Basis for a eigenspace (multiple choice problem)

The following (multiple choice) problem is from a test review. For the given matrix $A$, find a basis for the corresponding eigenspace for the given eigenvalue. $$A = \begin{bmatrix}1 & 6 ...
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25 views

Intuition behind Equation for point on a line between two solutions for linear equations

I am trying to solve some basic linear algebra problem. The book says a point on a line between two solutions ( for linear equations) is given by $c (sol1) + (1-c) (sol2)$ I don't understand this. ...
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37 views

Understanding Householder Transformations

I've been thinking about Householder transformation for the past few days and one point appears to be escaping my insight. I hope that the following description helps someone correct my understanding ...
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1answer
25 views

Find the maximum value of this form

Let $A,B$ be $n\times n$ real symmetric matrices such that $B$ is positive definite. Show that $G$ defined below attains a maximum value at an eigenvector related to $A$ and $B$. Also find the ...
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8answers
647 views

Why is this set of polynomials linearly dependent?

$$1 + 2t+ t^2, 3-9t^2,1 + 4t + 5t^2$$ (A) Linearly dependent or (B) Linearly independent The answer is A from the answer key. This is a test review. I don't see that either ...
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0answers
13 views

Lagrangian complement in symplectic vector space

Let $(V,\omega)$ be a vector space over $K$ together with symplectic form $\omega : V \times V \rightarrow K$. Let $U \subseteq V$ be a Lagrangian subspace (in other words, $U = U^{\perp}$). I want to ...
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3answers
30 views

Prove {$v_1,v_2,w$} is a basis for vector space _V_

A problem from my textbook states: Let {$v_1,v_2,v_3$} be a basis for a vector space $V$. Prove that, if $w$ is not in $sp(v_1,v_2)$, then {$v_1,v_2,w$} is also a basis for $V$. Assume ...
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1answer
35 views

Particular form of the kernel of a positive matrix

Let $\mathbf L$ be a positive ($L_{ij} \geqslant 0$) $k \times n$ matrix with $k < n$. I'm looking for a matrix $\mathbf H \in \mathbb R^{n \times (n-k)}$ with two properties: columns of ...
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4answers
60 views

If $V$ is a vector space, then, proving that…

I have a big problem with this problem... : If $V_m(\mathbb{R})$ is a vector space whose dimension is "$m$" then Proving that "$m$" is even number if and only if exist an endomorphism $J$ of ...
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1answer
16 views

Reducing a Matrix with Guass-Jordan elimination

I am doing some homework, but I am stuck on this problem. I have to take a (I'd upload an image, but being new, I can't): -x+y=-2 -3x+2z=13 2x-2z=-6 Here's a picture of my work(I can only link ...
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2answers
35 views

Problem: linear form

I need help with this problem: we are on $\mathbb{R}^2$ and we consider two non-null vectors, $ \vec{v}, \ \vec{w}$, and a non-trivial linear form $\phi\in{\mathbb{R}^2}^*$. Now if ...
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3answers
23 views

Problem about planes

Say we have $2x+3y+3z=0$ which is a plane. Does that plane have infinite dimensions (it is a 2D "object" — forgive me as I am not a mathematician — but each side has infinite length) or is it just the ...
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0answers
42 views

Are all 7-dimensional cross products isomorphic?

Let $\hspace{.04 in}f$ and $g$ be bilinear maps on $\mathbb{R}^7$ which satisfy the orthogonality and magnitude conditions. Does there necessarily exist a linear map $\: \phi : \mathbb{R}^7 \to ...
2
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2answers
32 views

Proving that $h$ is an automorphism if $2h^2+4h+2I=0$

I dont know how i can solve this problem. You have one vector space $V$ with a linear transformation: $h: V \to V$. So we have an endomorphism. The question is to show that if $h$ satisfies ...
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2answers
36 views

Distance between point and plane - why use the dot product?

So according to this, the signed distance between a point and a plane will be the dot product of the plane's normal vector (does it have to be a unit vector?) and the point-in-plane minus the point ...
2
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2answers
33 views

Dimension and basis of a quotient space

I'm having some problems understanding this: $$V = \mathbb R^3\text{ and }W = \{(x,y,z) \mid x+y+z=0\}$$ So I want $V/W$ and a basis to it. $$\dim V = 3$$ $$\dim W = 2$$ $$\dim V/W = 1$$ But a ...
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1answer
9 views

Find a specific intersection point of line and Fourier series using Newton–Raphson method when graphs have more then one intersection points

I need to find an intersection point of two graphs in polar coordinates. First is defined by a simple line $y=kx+b$. Second — by Fourier series $$ r=r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] ...
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0answers
17 views

What is pseudospectra of matrix polynomials? .

What is pseudo spectra of matrix polynomials? Please guide me with some example or some reference regarding it. Thank You!
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2answers
43 views

For what values of $a$, $b$, and $c$ the above system has: One solution. Infinitely many solutions. No solutions.

I am stuck with this now, I tried reducing the matrix to row echelon form, but it gets a bit hard. Is there not a simpler way? The system is: \begin{align*} a x + b y − 3 z &= −3\\ −2 x − b y + ...
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5answers
168 views

Find the dimension of a vector subspace

I'm doing a problem on finding the dimension of a linear subspace, more specifically if $\:$ {$f \in \mathcal P_n(\mathbf F): f(1)=0, f'(2)=0$} is a subspace of $P_n$, what is this dimension of ...
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1answer
34 views

Eigenvector and eigenvalue of the differential operator $L(x)=x''+3x'-4x$

This is a follow up question to this one. Just to summarize. I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x):=x''+3x'-4x$$ In other words I want to find ...
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0answers
13 views

Properties or solution of C=I+wCw^T matrix equation?

In a project, I came to the following matrix problem: $$C_1=wC$$ $$C=I+wCw^\dagger$$ Where the unknown matrices are $C$, which is hermitian positive definite, and $w$, general not hermitian, no ...
2
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3answers
134 views

Find a constant so matrix is invertible

I am doing some exercises from my Linear Algebra textbook and i have come across an exercise which I don't quite understand. Every exercise is graded with numbers from [1] to [5]. [1] is meant to be ...
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2answers
22 views

Span and linear independence. Test review; not homework.

The following is a question on my exam review. The answer (from the answer key) is C, but why? Here is my analysis: The set does not span $R^3$ because there is not a pivot in every row of ...
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1answer
37 views

What is the relation between rank of a matrix, its eigenvalues and eigenvectors

I am quite confused about this. I know that zero eigenvalue means that null space has non zero dimension. And that the rank of matrix is not the whole space. But is the number of distinct eigenvalues ...
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2answers
34 views

The set of all vectors of the form $(a-4b,5,4a+b,-a-b)$ is not a vector space

This isn't homework, it's test review. Will someone tell me if my analysis is correct? The answer is D (not a vector space) because the zero vector is not possible. Setting $a = 0$ and $b = 0$ makes ...
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1answer
50 views

Consequences of the positivity condition $v^t A v > 0$ for the eigenvalues of $A$

Let $A$ be an $n \times n$ symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive ...