Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
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195 views

What is the dual matrix (of a sample covariance matrix)?

Let $A$ be a matrix. I am most interested in the real, symmetric case, but for full understanding let's let $A$ be complex. What does it mean for $A^D$ to be the dual matrix of $A$? Can we interpret ...
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30 views

Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which ...
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53 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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44 views

What does the following matrix inequality mean?

Suppose $\{A_i\}_{i=1}^n$ is a collection of $m\times m$ matrices. I'm trying to understand the following criteria: There exists $\lambda\in[0,1)$ such that for all $x\in\mathbb{R}^m$ $$ \lambda ...
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99 views

Square root of positive definite nonsymmetric matrix

Let $N$ be a nilpotent matrix in $M_n({\mathbb R})$, such that $(I+N)^2$ is “positive definite” (but not necessarily symmetric) in the sense that $<X,(I+N)^2X>$ is positive for any nonzero ...
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42 views

Dynamical System of Matrices

Let $T_{nm}$ be the set of all possible binary rectangular matrices of dimension $n\times m$. The cardinality of $T_{nm}$ would be $2^{nm}$. Let f be a map from $T_{nm}$ to itself. Consider a ...
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43 views

Maximum and minimum value of $\|A\|_1$

Let $A$ be a $2\times 2$ real matrix. If $A$ is orthogonal, determine its maximum value and minimum value of $\|A\|_1$. My answer: Let $A=\begin{pmatrix} a \quad b\\ c \quad d \end{pmatrix}$. From ...
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50 views

How to prove that the determinant is the same no matter how you take it?

To find the determinant, pick a row and move along it creating minors and use the recursive definition of determinant. How do we know that the determinant will be the same no matter which row you ...
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64 views

Solving an equation with the form $Ax=b$

$$\begin{array}{l} \left( \begin{array}{l} \begin{array}{*{20}{c}} 1 & 2 & 3 & \cdots & n \\ \end{array} \\ \begin{array}{*{20}{c}} 2 & 3 & 4 & \cdots & ...
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On partition of matrix

Let $S \in \mathbb{R}^{n\times n}$ be positive definite matrix partitioned by $$S = \begin{pmatrix} S_{11} & S_{12} & S_{13}\\ S_{21} & S_{22} & S_{23}\\ S_{31} & S_{32} & ...
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69 views

Computing the decomposition of a representation of $S_n$

I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily? The best approach I have so far is as follows: Find a ...
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61 views

Relationships between Reduced Row Echelon Form and the Fundamental Four Subspaces [inspired by Strang P143 3.2.34]

I'm trying to apprehend all the links between two matrices' RREFs and their $4$ fundamental subspaces. Does $RREF(A) = RREF(B) $ $1.1.$ $\implies null(A) = null(B)$? True because $null(A) = ...
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59 views

Find the rank of the matrix

Let $X,Y\in\mathbb R^n$ be two non zero (column) vectors. Let $Y^T$ denote the transpose of Y. Let A = $X Y^T$. What is the rank of $A$ and what is the necessary and sufficient condition for the ...
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132 views

Jordan canonical forms determined by a minimal polynomial

Find the Jordan canonical forms of all $9\times 9$ matrices over $\mathbb{C}$ with minimal polynomial $x^2(x-3)^3$. My method: each factor of the minimal polynomial corresponds to a type of Jordan ...
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95 views

Difference between Householder Reflections and Gram-Schmidt?

In numerical QR decomposition, when we calculate the orthonormal factor Q of a matrix, what is the difference in results if we use Householder Reflections to normalize the matrix or use Gram-Schmidt ...
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81 views

Non Existence of matrices $A,B\in M_n(\mathbb{R})$ such that $(I-(AB-BA))^n=0$

Question is to Prove: Non Existence of matrices $A,B\in M_n(\mathbb{R})$ such that $(I-(AB-BA))^n=0$. This question has already been asked already but then i am asking for clarification of another ...
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206 views

Why doesn't every square matrix have $n$ linearly independent eigenvectors? [Strang P310 6.1.26]

Curt Solution: First reason: The nullspace and column space can overlap, so $\mathbf{x}$ could be in both. Second reason: There may not be $r$ independent eigenvectors in the column space. Longer ...
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56 views

pullback of density

So I have been reading some differential geometry and they are talking about density's and they claim that a pull back of a density is a density but I only have a partial proof of why this is true. ...
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dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that ...
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44 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
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106 views

Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
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55 views

Deducing that polynomials span

Let us say that we are dealing with a countable family of polynomials with real coefficients in $n$ indeterminates that commute. Are there any known/common nice systematic ways to tell if their span ...
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45 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
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76 views

Basis for an infinite dimensional vector space.

Is there any good paper that focus on the topic of basis for infinite dimensional vector space that I can read/ study. I found some papers that mention about this topic online, but they are very brief ...
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139 views

Determinant proof

Let $A\in M_n(\mathbb C)$ and $\alpha \in \mathbb C$. If $B$ is the matrix obtained by multiplying a single row of $A$ by $\alpha$, then det$(B)=$ $\alpha$ det$(A)$. I'm trying to understand and use ...
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100 views

Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
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47 views

Linearly (In)Dependency?

I really need some help with this proof.. Let V be a vector-space, $(A_i)_{i\in N}$ a sequence of linearly-independent subsets of V with properties: $ A_i \subseteq A_{i+1}$ (for all $in \in N$ ) ...
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1k views

Determinant of symmetric tridiagonal matrices

Given an $n\times n$ tridiagonal matrix $$A =\left(\begin{array}{ccccccc} ...
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Are solutions of $\frac{1}{2}(A^T+A)x=b$ and $Ax=b$ related?

I saw some statements about these 2 systems while I was reading something about linear algebra. So I am curious if the solutions of these 2 systems are related. If it is, how are they related? Thanks ...
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106 views

Eigenbasis and diagonal Matrix

Any transformation $T:V \rightarrow V$ can be cast into a diagonal matrix if there are $n$ distinct eigen-values for $T$, now it is said that $T$ becomes a diagonal matrix w.r.t. eigen-basis, does ...
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566 views

How can I solve system of linear equations over finite fields in WolframAlpha?

Is it possible to solve system of linear equations over finite fields using Wolfram Alpha? If yes, how can I do that? Let us take a system $x+y+z=0$, $2x+y+2z=0$, $x+3y+z=0$. If I want to solve this ...
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52 views

Basis in $L^2([-\pi,\pi])$

Consider the space $L^2([-\pi,\pi])$. Show that the functions $f_0(x)=1,f_1(x)=x,f_2(x)=x^2,\ldots $ form a basis. The functions are linearly independent (no linear combination adds up to zero). But ...
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120 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
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109 views

R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
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108 views

Constructing a directed graph from its spectrum

This is related to the following question from cs theory stack exchange: http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem So it seems as if given a sequence of real ...
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37 views

A work rate problem

Micheal paints $\frac{1}{p}$ of a building in 20 mins, what fraction of the same bulding can Hena paints in 20 mins, if they paint the building in an hour, working together. Answer options: ...
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67 views

How to quickly approximate the eigenvectors of a symmetric matrix

Given a symmetric $n \times n$ matrix $A$, is there any algorithm that can quickly approximate all of its eigenvectors? By "quickly", I mean with time complexity less than $\mathcal{O}(n^3)$.
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LU Factorisation - When are there 0s in L and U? (Strang P96 & P105 2.6.21)

Assume no row exchanges. When can we predict zeros in $L$ and $U$ ? $1.$ When a row of $A$ starts with zeros, so does that row of $L$. $2.$ When a column of $A$ starts with zeros, so does ...
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43 views

Linear Complex Structure and Kähler Angles

I am trying to read Donaldson's paper on symplectic submanifolds http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214459407 and am getting a bit ...
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38 views

The singularity of a family of matrices

Let $l\ge2$ be an even integer, $\zeta$ be a primitive $l$th root of unity in $\mathbb{C}$. Is it true for any $\alpha=(\alpha_1,\dots,\alpha_l)$ and $\beta=(\beta_1,\dots,\beta_l)$ such that ...
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Are periodic points dense in the unitary group?

In $U(1) = \{z \in \mathbb{C} : |z| = 1\}$, it is well known and easy to see that the set of $z$ so that $ z^n = 1 $ for some $n \in \mathbb{Z}_+$ are dense. Does this fact generalize to the group ...
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Problem involving subspaces and linear transformations

I'm asking for some opinions about my proof! $V$ and $W$ are vector spaces, and $T : V \rightarrow W$ is a linear transformation. $Z$ is a subspace of $W$, and $U$ is the set of all $\textbf{x} \in ...
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73 views

About scalar products on $\mathbb{R}^n$

In what follows, $\mathbb{L}^{n}$ is the Lorentz space (Euclidean space $\mathbb{E}^{n}$ with the Lorentz scalar product). Theorem Let $P$ be a $k$-dimensional subspace of $\mathbb{L}^{n}$. Then ...
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142 views

What can we say about two graphs if they have similar adjacency matrices?

Suppose we have two (finite, simple, undirected) graphs, what can we say about these graphs if they have similar adjacency matrices? Observations to begin with: If $G_1$ and $G_2$ are isomorphic, ...
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100 views

Mathematics Courses for an Economist

I am an Economist and I am interested in further developing my mathematical knowledge and skills. I would like to get your opinions on the topics that I should cover and which are also important for ...
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126 views

When every matrix is a sum of nilpotent (idempotent or invertible) matrices ??

Let $R$ be a ring with non-zero identity. Consider the following three properties: Every $A \in M_n(R)$ is a sum of nilpotent matrices. Every $A \in M_n(R)$ is a sum of idempotent matrices. Every ...
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601 views

Prove the solution of the differential equation C(y)=0 has a given form

Let $A$ and $B$ be two constant-coefficient operators whose characteristic polynomials have no roots in common. Then let $C=AB$. Prove that every solution of the differential equation $C(y) = 0$ has ...
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286 views

How can I show that two infinite-dimensional vector spaces are isomorphic?

How can I show that two infinite-dimensional vector spaces are isomorphic? Can I define a function which maps a basis of the vector space $V$ to a basis of the vector space $W$? And are there more ...
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61 views

Weil operator of elliptic curve

Let $V$ be a $1$-dimensional $\mathbb{C}$-vector space and $\Lambda \subset V$ be an elliptic curve (=lattice). Let $C : V \rightarrow V$ be the multiplication by $i$. Consider the two following ...