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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Determinant of M

How to find the determinant of the $n\times n$ matrix $M$, whose all the entries are zero except 1st row, 1st column and diagonal entries: $$M= \begin{bmatrix} -x & a_2 & a_3 & \cdots ...
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1answer
101 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in V\\ ...
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0answers
42 views

Transformation law for symmetric rank-2 tensors?

A rank-2 tensor $M_{ij}$ transforms as $M_{ij} \rightarrow O_{ik} O_{jl} M_{kl}$, where $O$ is some element of $SO(n)$. We can always get a symmetric tensor from $M_{ij}$ through $M_{ij}^s =M_{ij} + ...
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0answers
16 views

SVD for square matrix

I already know the concept of SVD applyed on an mxn matrix. Eigen vectors can't exist for a non-square matrix, but singular-vectors can. My question is: does SVD on a square matrix relate to ...
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0answers
34 views

How do Lie algebra elements act on symmetric and antisymmetric representations?

The Lie algebra of a group acts on itself through the commutator $ T_a \in ad$: $$ T_a \circ T_b = [T_a,T_b] \in ad $$ I assume the same should be true if we have an antisymmetric adjoint, as for ...
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9answers
203 views

Show that $B$ is invertible if $B=A^2-2A+2I$ and $A^3=2I$

If $A$ is $40\times 40$ matrix such that $A^3=2I$ show that $B$ is invertible where $B=A^2-2A+2I$. I tried to evaluate $B(A-I)$ , $B(A+I)$ , $B(A-2I)$ ... but I couldn't find anything.
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1answer
23 views

Finding a generating set of vectors

I want to solve the following task: Find the minimal generating set (german: "minimales Erzeugendensystem") for the set S: S = { $\begin{pmatrix} 1 \\ 1 \\ 0 \\1\\1 \end{pmatrix}$, ...
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1answer
21 views

How to determine positive or negative definite of a bordered Hessian ?

I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully ...
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0answers
12 views

Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function ...
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3answers
138 views

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. [on hold]

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. it is a question from a test i had yesterday and this is how it was ...
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0answers
9 views

Reducing a pair of indefinite quadratic forms to the canonical form

Assume $A, B$ being a pair of symmetric matrices over reals. Let $$ \varphi_1(x) = (x, Ax)\\ \varphi_2(x) = (x, Bx). $$ There's a well-known result that if $A > 0$ then the pair of forms can be ...
2
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1answer
584 views

Computing orthogonal projection onto range space of a given matrix

I have matrix $A =\begin{bmatrix} 2 & -2 & 0 & 0 & 0 & 0 \\ -2 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & -2 & 0 & 0 \\ 0 & 0 & -2 ...
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0answers
13 views

Can the transposition of an arbitrarily-sized matrix be broken up to smaller transpositions?

I'm working with binary matrices. Let's assume that I have an algorithm that is very efficient in transposing 8×8 or 8×16 matrices, but I would like to transpose matrices with an arbitrary size. ...
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4answers
134 views

Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
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1answer
41 views

If $A$ is an $m\times n$ matrix, $B$ is an $n\times m$ matrix and $n<m$, then $AB$ is not invertible.

The question was given in the early chapters of Linear Algebra by Hoffman & Kunze, so I am trying to give a proof with only the tools given to me so far - which are mainly row reduction and ...
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0answers
13 views

volume of the pyramid stub in points. [on hold]

Pyramid stub is in 3d coordinate system where coordinates to the points are A:(30,0,180) D:(0,0,180) E:(-15,-15,230) F:(45,-15,230) and H:(-15,45,230. Big square on top has points FEHG and small ...
2
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1answer
48 views

Derivations of important algebras?

After knowing the importance of studying derivations of an algebras(Why we wonder to know all derivations of an algebra?), this problem naturally raised "what is the space of all derivations of ...
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2answers
36 views

How to Find two points of the linear equation [on hold]

I need to find two points of the linear equation which I have given below: $$ 3x-6y=1 $$ I need to plot these points on the graph.so I need to find two points which can be easy to plot on the graph.
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1answer
16 views

Update PageRank given extra links

I have a stabilized importance vector $x_k$ that is the PageRank of a series of webpages as defined by the links between them. Graphically, this is the equivalent of a graph where nodes are pages and ...
2
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1answer
36 views

Show that $\{w_1,\dots,w_p,v_1,\dots,v_q\}$ is an orthogonal set and spans $\mathbb{R}^n$

These series of questions build up on each other i'm stucked on the last one, i'm also not sure if all of these work but I am pretty convinced they do. Let $W$ be a subspace of $\mathbb{R}^n$ with an ...
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3answers
31 views

Dimensions of a basis of a coordinate space

I need a little clarification on the relationship between the basis, its dimension and their corresponding real coordinate space. Suppose we are operating in the fourth coordinate space ...
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2answers
62 views
+50

Asymptotic series of a matrix-valued function.

Consider the following matrix $$f(\lambda)=\left( \frac{\lambda-1}{\lambda + 1} \right)^{\nu \sigma_3} \ \ \ \lambda \in \mathbb{C} \setminus [-1,1]$$ where $\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 ...
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2answers
29 views

What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
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3answers
30 views
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0answers
10 views

Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
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1answer
25 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
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2answers
28 views

Orthonormality of the columns of a matrix

I am studying orthogonal columns and matrices right now and I have encountered the following theorem: Theorem An $m \times n$ matrix $U$ has orthonormal columns if and only if $U^T U = 1$. Is it ...
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1answer
19 views

Help on Solutions to Systems of Equations

Here is a screenshot: http://imgur.com/gallery/Wh6ksgO/new I was looking at my Linear Algebra quiz solutions and I saw the following: "Thus from RREF, we can see the system if consistent and contains ...
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1answer
16 views

Commutation of a partial trace with an operator

Let the partial trace $\mathrm{tr}_B$ be a mapping from an endomorphism End$\left( H_A\otimes H_B \right)$ onto an endomorphism End$\left( H_A \right)$. Then the partial trace is defined as $$ ...
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1answer
19 views

Given line $e$ and plane $\alpha$, find all points $Q$ on $e$ such $d(Q, P)= d(Q, \alpha)$

Can someone help me with this question and show my step by step process. I am unable to solve it. Thank you. $P(4,2,5)$ The plane $\alpha$ is given by $2x+y-2z=2$ The line $e$ is given by ...
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2answers
39 views

What is switching rows useful for?

I've learned about elementary row operations, there is one of them that seems a little bit weird to me: The row switching. It seems that a system of equations: $$\begin{eqnarray*} ...
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1answer
25 views

Learning to solve complex inequalities in many variables

below is a very specific inequality problem. I would like to know how to solve it so I can apply it to more complex problems. The equations are as follows: $$3.5x−2.5y−3z=A$$ $$−7.5x+3.75y+5.25z=B$$ ...
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0answers
19 views

Bound on the difference of matrix diagonals

I have two diagonal matrices $\Lambda,\hat{\Lambda}\in\mathbb{R}^{n\times n}$ with non-negative diagonal elements. And I have two matrices $W,\hat{W}\in\mathbb{R}^{m\times n}$, with $m\geq n$, each ...
20
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3answers
656 views

Old AMM problem

I am working on an old AMM problem: Suppose $A,B$ are $n\times n$ real symmetric matrices with $\operatorname{tr} ((A+B)^k)= \operatorname{tr}(A^k) + \operatorname{tr}(B^k) $ for every positive ...
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0answers
14 views

matrix with fractional exponent, not getting expected output in Matlab/Octave

I have a matrix exponential function that is called a number of times in an integration routine from the heat conduction model I'm trying to implement. It works, and my results match the samples in ...
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1answer
35 views

Given an area, calculate the angle of a wedge out of an annulus between a square and a circle

If we have a shape similar to this picture: Where the square length is less or equal to the circle's diameter, then I believe the term for the blue area is the annulus. I was wondering if it is ...
2
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1answer
70 views

Find integer solutions equation of ${ x }_{ 1 }^{ 4 }+{ { x }_{ 2 }^{ 4 }+ }{ x }_{ 3 }^{ 4 }+…+{ x }_{ 14 }^{ 4 }=1599 $

I tried to solve this equation,but can't end up $${ x }_{ 1 }^{ 4 }+{ { x }_{ 2 }^{ 4 }+ }{ x }_{ 3 }^{ 4 }+...+{ x }_{ 14 }^{ 4 }=1599$$ My work: Consider arbitrary $x_{ i }=2k,\quad \forall ...
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0answers
13 views

Simulate ICA Source Signal

I am using the fastICA package in R for a matrix of time series information. However, if I wanted to simulate the process for risk management purposes how exactly could I do this? For example lets ...
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0answers
16 views

System of linear equations and Fredholm's alternative

I am learning linear algebra and bought the book from Gilbert Strange: Introduction to linear algebra and trying to understand the four fundamental subspaces. I know that a system is solvable if b is ...
0
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1answer
19 views

A simple proof for angle inequality in inner product spaces

I am looking for a smiple proof for the following fact: Let $u,v,w$ be vectors in an inner product space $V$. Then it holds: $\theta (u, v)≤\theta(u, w) + \theta(w, v)$ (Of course if they are all in ...
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1answer
50 views

Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
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4answers
84 views

How to prove rigorously that you need $m \geq n$ equations with $n$ unknowns to be able to solve a system of equations?

How to show that you need $m \geq n$ equations with $n$ unknowns to be able to solve a system of equations? I mean it seems obvious to me that you can't solve e.g. 2 equations with 3 unknowns, but how ...
3
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2answers
51 views

Finding a basis of a complex vector space over $\Bbb R$ given a basis over $\Bbb C$

Suppose $X$ is a vector space over $\mathbb C$ and has as basis $\{e_1,e_2,\ldots,e_n\}$. Now regard $X$ as a vector space over $\mathbb R$. What will be the basis? My thoughts: I considered ...
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1answer
33 views

Special Properties of Real Matrices With Real Distinct Eigenvalues

Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do ...
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1answer
40 views

General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where $$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$ or in other words, M commutes with all members of the the ...
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1answer
35 views

More on linear algebra vector subspaces

I am continuing on my journey of trying to understand vector subspaces. Question: Let $F(-\infty,\infty)$ be the set of all real-value functions defined at each x in the interval $(-\infty,\infty)$. ...
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9 views

Lipschitz continuity of invariant subspaces for parametrized matrices

Let $A(t)$ be a one-dimensional parametrized family of linear operators on $\mathbb{R}^m$ that has smooth dependence on $t$. Let $V_0\subset \mathbb{R}^n$ be an $n$-dimensional invariant subspace for ...
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3answers
73 views

Does every invertible complex matrix have an eigenvector?

Over $\mathbb{C}$ does every invertible matrix have at least one non-zero eigenvalue and an eigenvector? I'm generally confused about eigenvectors and eigenvalues. I understand that eigenvectors are ...
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1answer
55 views

Is an infinite system of (linear) equations solvable if all finite subsystems are?

I was wondering about the following. Let $A$ be an abelian group, $a_i$ variables indexed with some arbitrary set $I$ and assume we have an infinite set $E$ of linear equations in finitely many ...
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0answers
26 views

Signature of a complex matrix

If $A\in M_n(\Bbb R)$, we define its signature as the triple $(s^+,s^-,s^0)$, where $s^+,s^-,s^0\in\Bbb Z_{\ge0}$ denotes respectively the number of eigenvalues $>0,<0,=0$. How can we define ...