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Questions tagged [linear-algebra]

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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Is tr(TA)=tr(TB) for any Hermitian operator T equivalent to A=B? A,B is operator.

I wonder whether this is true: For any Hermitian operator $T$, tr($TA$)=tr($TB$), this is equivalent to $A=B$($A,B$ is operator). I tried to prove it by contradiction, but failed to prove: If $...
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4answers
9 views

Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$

Let $M_{n\times n}(\Bbb{R}).$ We say that $A$ is positive if $\langle Ax, x \rangle>0$ for $x\neq 0.$ I want to prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, ...
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1answer
42 views

Given the solution of $Ax=b$ for some $b$ finding the nullspace.

So i have 2 questions. First of all, we know that everything of the form $p+$ nullspace vector is a solution to $Ax=b$ when $Ap=b$. Now my first question is, how do we know these are all the solutions....
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15 views

When are almost block companion matrices which yield a given characteristic polynomial connected?

This is motivated by Are matrices which yield a given characteristic polynomial and have specified structure connected? Let $\mathcal E \in M_n(\mathbb R)$ be a subset with following form: we first ...
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0answers
18 views

Complexifying a Real Linear Operator

I'm preparing for some exams by studying previous exams and I think I have a solution but a few steps may be invalid. "Let $V$ be a real vector space of finite dimension $d > 2$. Let $T$ be a ...
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3answers
73 views

Are two vectors equal if their inner product with each vector from some generator set is the same?

I need to decide wether the following is true (and, of course, justify it): given $V = [v_1, v_2, ..., v_n]$, $v$, $w$ in $V$ and $<v,v_i> = <w,v_i>$, for any $i$, so $v = w$. I wrote the ...
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2answers
46 views

$F \circ G$ self-adjoint $\iff F \circ G = G \circ F$

Let $V$ be an finite inner product space and $F$ and $G$ two self-adjoint endomorphisms on $V$. Show that iff $F \circ G$ is self-adjoint, $F \circ G = G \circ F$. My Proof: "$\implies$": If $F \...
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0answers
19 views

Infering the shape of function from its components

Is it legitimate to say that, if all of the components of a function are linear (say of a vector function), then so is this function? How far can we take similair arguments without rigorous proof?
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1answer
18 views

Check My Proof - $F(v_j) = w_j$ is orthogonal

Let V be an inner product space and $(v_1, \ldots, v_n)$ and $(w_1, \ldots, w_n)$ be orthonormal bases of $V$. Let $F \in$ End$(V)$ be the distinct mapping with $F(v_j) = w_j$ for alle $j \in \{1, \...
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1answer
35 views

Represent the following system of equation in the Y = AX + B form: [on hold]

$$y_1 = 4x_1 + x_2 + x_3 + 9x_4 + 5$$ $$y_2 = x_1 - x_2 + 6x_3 - x_4 + 1$$ $$y_3 = -2x_1 + 3x_2 - x_3 + 2x_4 - 19$$ What do we need to do? Should we take all on one side and keep the 5 on the other ...
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3answers
42 views

Check My Proof - $V = \sum_{i = 1}^{n} U_i \implies V = \bigoplus_{i = 1}^{n} U_i$ if $U_i \bot U_j \ \forall i,j \in \{1, \ldots, n\}$

I am not sure, if my following proof is correct: Let $V$ be an inner product space. Prove that if $V$ is the sum of pairwise orthogonal subspaces $U_1, \ldots U_n$, the sum must be direct; $V = \...
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1answer
27 views

Injectivitity of $A-\mu I$ on generalized space

$V$ is a $K$-vector space. Given an eigenvalue $\lambda$, define $B_\lambda=\{v\in V|(A-\lambda I)^kv=0 \text{ for some }k \in \mathbb{Z}\}$ which is a subspace of $V$. Show that $A-\mu I$ is ...
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18 views

A bilinear mapping on a tensor product-explanation request

Let $X, Y$ be linear spaces over $K$, and $S,T:X\times Y \rightarrow K$ be bilinear forms. We put for $x,u \in X$, $y,v\in Y$: $ f(x \otimes u, y\otimes v)=S(x,y)T(u,v) $ and we extend $f$ on ($X\...
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2answers
45 views

Conditions for a matrix to have non-repeated eigenvalues

I am wondering if anybody knows any reference/idea that can be used to adress the following seemingly simple question "Is there any set of conditions so that all the eigenvalues of a real positive ...
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1answer
28 views

Eigenvalue of normal matrices

$A$ is a (complex) normal matrix, i.e. $AA^*=A^*A$ and if $Av=\lambda v$, show that $A^* v=\overline{\lambda}v$. Is there any direct and direct proof which does not involve the fact that it is ...
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2answers
33 views

Geometric visualization for volume of parallelepiped

Volume of parallelepiped defined by vectors \begin{align} \begin{bmatrix}a\\b\\c\end{bmatrix}&,\quad \begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix},\quad\begin{bmatrix}w_1\\w_2\\w_3\end{bmatrix} = \...
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0answers
22 views

Isometry of quadratic space

Every element of $K$ is a square if and only if every 2-dimensional form over $K$ is isotropic. In fact, if $\langle -1, d \rangle$ is isotropic, then $\langle - 1,d \rangle \cong \langle -1, 1\rangle$...
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1answer
56 views

Vector Field analog of Functions

Bear in mind that I don't have a formal education in mathematics. If functions form an abstract vector space then a single function can be considered as a member of a vector space. Then how can we ...
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1answer
34 views

transpose(u) * Matrix * u

I've seen this a couple of times during a machine learning lecture, for example in context of LDA, when looking at the Fisher Criterion. It can be expressed in two ways: $$J(w) = \frac{(m_1 - m_2)^2}{...
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1answer
29 views

How are these cross-product summations equivalent?

Trying to determine how the $X_{i+1}$ is no longer applicable by changing summation bounds: $$\sum_{i=0}^{n-1} (X_{i} + X_{i+1})(Y_{i+1} - Y_{i}) = \sum_{i=1}^{n} X_{i}(Y_{i+1} - Y_{i - 1})$$ Can ...
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1answer
44 views

geometric intuition behind an n-dimensional rotation matrix

How do I derive an n-dimensional rotation matrix from a geometric perspective? I have read on wikipedia that it preserves distance so that $Q^TQ = I$ but the explanation to be honest isn't very clear. ...
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1answer
29 views

Is this matrices exist, with this propertie

if $\begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix}$, $\begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$ $\in R(A)$, and $\begin{bmatrix} 2\\ 5 \end{bmatrix}$, $\begin{...
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1answer
23 views

Matrices that have the same fundamental subspace

If matrices $A$ and $B$ have the same fundamental subspace than $A=cB$ where $c$ is some scalar. I think it is true, because I try to disprove with this two matrices, they have the same dimension of ...
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4answers
65 views

can I perform 3 row operation at the same time in a 3 by 3 determinant? [on hold]

I mean is it necessary to keep one row unchanged. r1'=r1+r3 r2'=r2-r3 r3'=r3+r2. I did these operations and the value of determinant is unchanged.is this a valid operation.
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1answer
20 views

Euclidean space and set $M\cap M^{\bot}$

If $M$ is subspace of some Euclidean space $E$ than set $M\cap M^{\bot}$ sometimes is empty, sometimes is not. I read in book that $M\cap M^{\bot}=\{0\}$ so it can not be sometime empty sometimes not,...
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1answer
14 views

If $T\in \mathcal L(\mathbb R^n,\mathbb R^n)$ invertible, why are there orthogonale matrices $P,Q$ s.t. $T=PDQ$ withe $D$ diagonal?

Let $T\in \mathcal L(\mathbb R^n,\mathbb R^n)$ invertible. Why are there matrices $P,Q\in \mathcal L(\mathbb R^n,\mathbb R^n)$ orthogonal s.t. $$T=PDQ$$ with $D=diag[t_1,...,t_n]$ diagonale and $|\det ...
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1answer
27 views

Euclidean space and subspace

I have some question that I get for homework. Let $U$ and $W$ subspace of Euclidean space $E$. Then if $U$ is orthogonal on $W$, then $U^{\bot}$ is orthogonal on $W^{\bot}$. For second question I ...
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1answer
20 views

On proper notation of set of solutions for a linear equation system

For an endless set of solutions of a system of linear equations, one possible representation is a specific solution plus any of possible solutions of the related homogeneous system. My prof and tutor ...
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1answer
32 views

Finding the amount partitions with gives sizes of a multiset

A multiset $A$ contains $E$ positive integers. The multiplicity of each element is $r_i \; i=1,\ldots ,N$. $A$ is partitioned in $M$ (we do not necessary have $M=N$) ordinary sets (where elements are ...
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1answer
31 views

If differential operators are linear operators, what might it mean to act a differential operator to a function to its left?

Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an ...
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3answers
35 views

Finding matrices which P is null space

Let P is set all vectors $(x_1,x_2,x_3,x_4) \in R^{4}$ for which $x_1 + x_2 + x_3 + x_4=0$ a) Find one base of subspace $P^{\bot}$ b) Prove that P is subspace of $R^{4}$ then construct matrices ...
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0answers
18 views

Nonnegative irreducible matrices, Perron eigenvector and Dietzenbacher's theorem

Let $A\geq 0$ irreducible, and let $\hat{A}\geq 0$ and irreducible obtained by changing some entries of $A$. Consider $Ax=\rho x$ and $\hat{A}\hat{x}=\hat{\rho}\hat{x}$ the Perron's eigencouples, and ...
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0answers
14 views

linear operators problem 2

Let $V$ be a finite dimensional vector space,and let $\mathscr L(V)$ be the space of all linear operators on $V$. Suppose $f$ is a linear functional on $\mathscr L(V)$ such that $f(TS)=f(ST)$ for ...
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1answer
10 views

Similarity of matrix over field extension [duplicate]

Let $E/F$ be a field extension and let $A, B$ be $n\times n$ matrices over $F$. Assume that $A, B$ are similar over $E$, i.e. there exists $P\in \mathrm{GL}_{n}(E)$ s.t. $B = PAP^{-1}$. Are $A, B$ ...
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3answers
48 views

Show that differentiation $D = \frac d{dt}$ is a linear operator on the 3-dimensional space of ‘quasi-polynomials’ $e^t (a_2t^2 + a_1t + a_0)$.

Show that differentiation $D = \frac d{dt}$ is a linear operator on the 3-dimensional space of ‘quasi-polynomials’ $e^t (a_2t^2 + a_1t + a_0)$. Im a little unsure how to approach this: Using $(e^t, ...
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2answers
49 views

find the singular value and eigenvalue of A? [on hold]

find the singular value of the $ n \times n $ real symmetrics matrices . $$A=\left(\begin{matrix} 1 & 1 & ... & 1\\ 1 & -1 & ... & 0 \\ .& .& . & .\\ ...
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0answers
30 views

Integration over portion of circle

I need to find the integral region $\mathcal R$ which covers area CPRQ portion of the circle radius $t$. The integration looks as $$I=\int\int_{\mathcal R}f(r,\theta)dr d\theta$$ Can someone help ...
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Literature on BFGS

so I have some questions and am looking for some papers to read for my own knowledge and understanding. So one may think of BFGS as poor man Newton method in terms of convergence, requires less but it ...
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1answer
14 views

Unimodular symmetric integral matrices with diagonal 0 and no $\pm 1$ entries

I am wondering if there exist any square symmetric matrices $A$ with integer entries, all zeros along the diagonal, determinant $1$, and the property that none of the entries in the matrix are equal ...
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1answer
38 views

If $AB$ invertible $\implies$ $A,B$ invertible, Given that $A,B\in M_{n\times n}(F)$

I know how to prove if $A,B$ invertible then $AB$ is invertible and $(AB)^{-1}=B^{-1}A^{-1}.$ But I'm not sure about how to prove the converse. My attempt: If $AB$ is invertible, then $B$ must be ...
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0answers
14 views

How to flatten out a curve based on the dot product to be linear?

I am interpreting a value based on the amount an object is "facing" a wall in a performance critical computer simulation. I take the dot product between the object's forward vector and the negated ...
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0answers
19 views

Determining a polyhedron from a series of inequalities

Given the following equations: $x+y+2z \leq 8$ $y+6z \leq 12$ $z \leq 4$ $y \leq 6$ $x,y,z \geq 0$ we have to determine all the vertices of the polyhedron in $R^3$, and indicate which vertices ...
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3answers
81 views

Find the matrix X such that $A\cdot X = X\cdot A^T$

I am working at some linear transformations and I need to know if it is possible given a square real matrix $A$, to find a real square invertible matrix $X$ such that $$ X^{-1} \cdot A \cdot X = A^T$$...
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0answers
56 views

Solving underdetermined linear system using least squares

As in case of linear overdetermined system of equations, we can prove that the cost function i.e. the least square function is convex. But in linear underdetermined system, we know that there exist ...
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1answer
18 views

Show that if $F(x)$ is a polynomial over $K$, then $F(A)=0$ if and only if $F(T_A)=0$.

Let $V$ be the space of $n\times n$ matrices over a field $K$. Let $A$ be an $n\times n$ matrix. Let $T_A: V\to V$ be the linear transformation given by $T_A(B)=AB$. Show that if $F(x)$ is a ...
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1answer
30 views

The relationship between spectral decomposition / eigendecomposition and projection operators

I am trying to clarify the relationship between the spectral decomposition / eigendecomposition of a matrix and projection operators. I understand that there is a connection between diagonalizability ...
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0answers
27 views

Let $A\in M_{m\times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:

Let $A\in \mathbb{M}_{m\times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below: a)The map $\mathbb{R}^n\to \mathbb{R}^m$ given by $v\to Av$ is injective . b)There exists ...
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1answer
37 views

Derivation of formula to calculate the angle of inclination

I understand that in order to find the angle between two lines, you can calculate the angle of inclination of each of the lines and take the difference between those two. However, I'm not sure exactly ...
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0answers
42 views

Do automorphisms form a vector subspace? [on hold]

We have a vector space $V$. Then is the set of automorphisms on $V$ closed under addition and composition—i.e., does it form a vector subspace on the space of all linear transformations to $V$?
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22 views

Application of Higher Order Derivatives

If $V = \mathbb{R}^{n}$ and $W = \mathbb{R}^{m}$, and $f:V\rightarrow W$ is smooth, then higher order derivatives at $x\in V$, $D^{n}_{x}f$ can be thought of as symmetric, multilinear functions $V\...