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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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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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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34 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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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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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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33 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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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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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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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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110 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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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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230 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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59 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 ...
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What is mass frequency and how it arose

Gaussian-profile initial condition has the solution, $$\phi (r,t)=\frac{R^{3}}{2}\frac{A}{\sqrt{\pi }}\int_{0}^{\infty }ke^{-R^{2}k^{2}/4}\frac{\sin (kr)}{r}\cos (\omega t)\ dk,$$ From this equation ...
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If $f$ and $g$ are immersions, show that $f \times g$ is.

Is this proof correct? I am particularly uncertain with the last step. Consider $f: X \to Y, g: M \to N$. $\forall x \in X, df_x: T_x(X) \to T_y(Y)$ is injective. Similarly, ...
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347 views

Dot diagram (aka Young diagram) method for finding a rational cannonical basis

I am trying to use the method described in Friedberg, Insel and Spence's Linear Algebra textbook to solve the following problem (cf. Freidberg,Insel and Spence pp 534-545 4th ed.). The Authors call ...
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Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
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How to solve a system of 2 equations in this form

Equation1 => $Dxy-Zx+Fx^2+Gy^2-Hy+K=0$ Equation2 => $Mxy-Nx+Ox^2+Py^2-Qy+R=0$ These equations when solve will give at most 4 points of Intersection But how to solve these two equations. The above ...
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Existence of weights of a finite dimensional representation of a semisimple Lie algebra

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. I want to show that every finite dimensional irreducible representation of $\mathfrak{g}$ is a weight module, and I need the existence of at ...
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The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
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Converting a linear program into standard form

In especially, I have a question about the demand that if I have $ Ax \leq b$, then I can convert this into $A'x'=b$ for some new $A'$ and $x'$. I have given the system of equations: $20x_1+30x_2 ...
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Inner spaces - A question about adjoints.

Question: Let $V$ be a complex vector space over $\mathbb{C}$ with inner product $\langle ,\rangle$. Let $E$ be an linear operator on $V$ such that $E^2=E$ with adjoint $E^*$. Show that $E$ is ...
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74 views

About linear recurrence sequences

Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy \begin{eqnarray*} &&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\ ...
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329 views

Real projective space is Hausdorff

I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ...
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69 views

Extending transvections/generating the symplectic group

The context is showing that the symplectic group is generated by symplectic transvections. At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the ...
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Linear Algebra Terminology

I'm trying to translate a linear algebra term from my native tongue to English. The term refers to a set of $n$ vectors over $\mathbb{R}^d$ such that each $d$ vectors from the set are linearly ...
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Euclidean division of $X^n$

For $n\ge 2$ the euclidean division of $X^n$ by $f(X)$ (of degree 2) will be $$X^n=f(X)q(X)+\alpha_1X+\alpha_2\tag{$*$}$$ My question is when for some matrix $A$ we are given that $f(A)=0$ would a ...
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When is a matrix congruent to a diagonal matrix and how to find the congruent transformation?

What matrix can be congruent to a diagonal matrix and how can we find the congruent transform and the diagonal matrix? One special case is when the congruence is also similarity. For example, for a ...
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127 views

A basis of the symmetric power consisting of powers

Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It ...
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symmetric matrix

Consider the basis $β=\{(1,1,0),(1,0,-1),(2,1,0)\}$ for $\mathbb R^3$ Does the following matrix $A=[T]_β^β$ define symmetric mappings of $\mathbb R^3$? \begin{bmatrix}-1 & 1 & 2 \\ 1 & 4 ...
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Counting symmetric unitary matrices with elements of equal magnitude

Let $X$ be an $n\times n$ symmetric unitary matrix with elements of equal magnitude and the elements of the first row (and the first column, of course) are $1/\sqrt{n}$, i.e. $X_{j,k} = e^{i ...
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Computing the trace of a linear map, expressed in terms of a generating set which is not a basis

Assume that $V$ is a finite-dimensional, complex vector space such that $v_1,\ldots,v_n\in V$ generate $V$, but $\dim(V)<n$. Let $\phi:V\to V$ be a linear map which I can describe in terms of the ...
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positive semi-definiteness by functions

$\textbf{My Question:}$ We have a matrix $A$ whose elements are $A_{ij} = \Phi(\max(x_i+x_j-1,0)) $ where we have $ 0 \leq x_i \leq 1$ for $ i=1,2,...n$. I need to find the set of functions ...
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How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?

The question is pretty straight-forward: how do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$ ?
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Unitary space, inner product

Prove that the space of all complex-valued functions continuous on an interval $[a, b]$ becomes a unitary space if we define an inner product by the formula $$\langle f,g\rangle = ...
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313 views

Geometric interpretations of matrix inverses

Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through ...
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Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?

The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element is not adjacent to the diagonal, then there are ...
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Trace Minimization of Covariance Matrix

Given a matrix X whose rows contain observations collected at some locations. Can someone explain how trace minimization of covariance matrix $XX^T$ can lead to orthogonal / mutually independent ...
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how to use determinant calculate the area

fiend the area of the pentagon of the five vertices $(1,2),(4,1),(5,3),(3,7),(2,6)$, please use the way of using determinant. My idea is to cut the pentagon into some triangles, then calculate each ...
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What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
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finding the largest $p$ components of $x$

Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations: ...
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286 views

What is a good metric to compare matrices?

I have a matrix that I obtained from theoretical computation and I have another matrix which I obtained by actual data collection. How do I compare the two matrices? How do I state that one matrix is ...
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167 views

linear algebra approach to discrete cosine transform

I understand that the discrete Fourier transform simply changes basis to the discrete Fourier basis, which is an orthonormal basis of eigenvectors for any shift-invariant linear operator on $\mathbb ...
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Curvature of particular Riemannian metric

Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian ...
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Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
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774 views

Good introductory book for matrix calculus

Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
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What conditions must the constants satisfy so that each of these systems has a solution?

I'm attempting to teach myself linear algebra using this book http://joshua.smcvt.edu/linearalgebra/book.pdf One of the exercises is: ...