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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How can I recover the weights of a Laspeyres index number?

I'm trying to recover the weights used to compute a Laspeyres quantity index. The index number's formula is: $$ Q=\frac{\sum_{i=1}^{N}{p_i^0 q_i^t}}{\sum_{i=1}^{N}{p_i^0 q_i^0}} $$ where in this ...
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361 views

Real and Complex Canonical Form

I have to convert some quadratic forms into real and complex canonical forms. One of these forms is as such: $$q_1\begin{pmatrix} x \\ y \\ z \end{pmatrix} = x^2 +3y^2 +z^2 +2xy−2xz−2yz$$ From the ...
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39 views

Determine whether a system of equations implies the equality of two variables

I have a system of equations of the form $a_i\alpha+b_i\beta+c_i\gamma+d_i\delta=2$. All coefficients are positive integers, and I also have the extra restriction that ...
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83 views

Shilov Linear Algebra, Coordinate transformations

Why does he first introduce the transposed version of the transformation matrices? That is, the first column represents the first basis vector in terms of the old basis, not the first row. Then he ...
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84 views

Applying shur's lemma to make triangular matirx

$A=\begin{pmatrix} 2 & -1 \\ 1 & 0 \end{pmatrix}$ has eigenvalue $\lambda=1$. The eigenvector is (1, 1). To make triangular matrix, $U^{-1}AU=\begin{pmatrix} \frac{1}{\sqrt{2}} & ...
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115 views

Vector fields generating a transformation

It would be great if someone can explain to me what the following means: Vector fields $V_i, i=1,2,3$ generate 3 single-parameter groups of transformations in $\mathbb R$ -- $$\tilde x ...
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40 views

Transformation, matrices

What am I missing here? Let $A,B$ be matrix-valued functions of $(x,y)$. Given that $$A_x-B_y+AB-BA=0$$ where the subscript denotes partial differentiation w.r.t. that variable, I wish to show ...
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321 views

Orthogonal complement in an infinite dimensional vector space.

Let $V$ be an infinite dimensional vector space and $A$ be a subspace of $V$. Is there always an orthogonal complement of $A$ in $V$? If not, is there a counter-example? Thank you very much.
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502 views

Gaussian elimination mod k

We have this assignment in programming class, but I believe posting it in math will make more sense. So we're supposed to write a program that takes $n$ equations with each $n$ coefficients, ...
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164 views

General definition of a line

In the book on Linear Algebra that I am using, the author defines a line in an arbitrary vector space $V$, given direction $ 0 \neq d \in V $ and passing through $ p \in V$ as $ l(p;d)= \lbrace v\in ...
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111 views

Time-derivative of an operator

Would I be right in thinking that the operator $$\hat O'(t)$$ is different from the operator $$D\hat O(t)$$ where $D={d\over dt}$, since when acting on a function $f$, the second corresponds to $$\hat ...
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150 views

Question about the elementary divisors of a special matrix

I have the following question: Is there a closed formula for the elementary divisors of the Matrix $M={(m_{ij})}_{i=1,...,n,\ j=1,...,k}$, where ${m}_{ij}$ is the greates common divisor of $i$ and ...
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99 views

Rank-nullity theorem and binary codes

I am asked to prove the fact that if $C$ is an $[N,k]$ code, and $C^{\perp} = x \in \mathbb{F}_2^N$ $|$ $(x,c) = 0$  $\forall c \in C$, then $\dim C + \dim C^{\perp} = N$. I am regrettably far ...
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29 views

Bibliographic References on the number of invariant subspaces of a linear transformation

I need bibliographic references on the number of invariant subspaces of a linear transformation. Is there any application for counting the invariant subspaces?
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128 views

Show that its a Generalized Eigenvalue problem

Show that the minimizer is obtained by a generalized eigenvalue problem. $$\alpha=\underset{1^TK\alpha=0; \ \alpha^TK^2\alpha=1}{\text {arg min}} \gamma ||f||_{K}^2+f^TLf$$ Details: $K$ ...
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125 views

Would this be bounded?

Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of $M$ is less than $1$. Let $I_{r}$ be an $m$ ...
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99 views

Smallest field of rationality of a subspace of a linear space

The following definitions and proposition are due to Bourbaki's Algebra Ch. II. Let $K$ be an extension of a field $k$. Definition 1 Let $V$ be a vector space over $K$. Let $V_0$ be a $k$-subspace ...
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When is a vector-valued function additively separable?

Suppose the map $u:(\mathbb{R}^{|A|})^{n} \rightarrow \mathbb{R}^{|A|}$ can be written in an additive form, i.e. there exist real-valued functions $g_{i}$ s.t. $u(x_{1},\dots,x_{n})=\sum ...
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66 views

Optimized Algorithm for Distance Matrix Solution

I've been looking for an optimized algorithm for solving a distance matrix (a hollow, skew symmetric matrix), but I haven't been able to find anything but papers discussing repopulating sparse ...
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65 views

linear equation system question

I have a set of linear equations which follow this shape: ...
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176 views

Simultaneous Eigenvalue Problem

I have what I think is a simultaneous eigenvalue problem in three parameters: $$\alpha A_1x + \beta B_1x + \gamma C_1x + D_1x = 0$$ $$\alpha A_2x + \beta B_2x + \gamma C_2x + D_2x = 0$$ $$\alpha A_3x ...
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95 views

Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
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51 views

Eigen values of a certain type of block matrix

Consider a $N \times N$ hermitian matrix $A$. Consider a complex $N \times 1$ vector $b$ and positive constant $c$. Given $A$ (hence its eigen-values), can we find the eigen-values of the following ...
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168 views

An old linear transformations homework problem

Here's a homework problem I never got around to doing when I took the course. I'll state it in full and share what what I know and where I'm confused (more in the latter category than the former I'm ...
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51 views

Algorithms for Performing Large Integer Matrix Operations w/ Numerical Stability

I'm looking for a library that performs matrix operations on large sparse matrices w/o sacrificing numerical stability. Matrices will be 1000+ by 1000+ and values of the matrix will be between 0 and ...
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93 views

Matrix over a ring with given kernel

Let $R$ be a (commutative, unital,) Noetherian ring and let $M$ be an $m\times n$ matrix over $R$. The columns of $M$ span a submodule $\widetilde M$ of $R^m$, but in general $\widetilde M$ will not ...
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66 views

Polynomial-time algorithm to check if a vector intersects a hypercube

I am near the end of a long research problem which was formulated in terms of graph theory to solve a problem in quantum error correcting codes, and I have now constructed some machinery using linear ...
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102 views

Norms on Dual Spaces

Suppose that $\varphi$ is a norm on $\mathbb{R}^n$ such that the set $$\varphi_1 = \{x \in \mathbb{R}^n : \varphi(x) = 1 \}$$ is a polyhedron. Let the dual norm $\varphi^*$ be defined as usual: ...
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57 views

Maximum value for parameter

I am facing the following problem: A number of a adults, b children older than 12, and c children younger than twelve attend an event. The sum of all people a+b+c=100. The prices are \$6 per adult, ...
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83 views

what is the pseudo-inverse of this type of matrix?

what is the pseudo-inverse of this type of matrix $$ A=\left[\begin{array} {C}a_0&0&b_1&b_2&\ldots&b_n \\ a_1&b_n&0&b_1&\ldots&b_{n-1} \\ a_2 ...
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120 views

Change of variables problem

I came across this problem yesterday where i wanted to change variables in an integral like below. $$\iiint f\left(x_{1},x_{2},x_{3}\right) dx_{1}dx_{2}dx_{3}\tag{1}$$ so $y_1 = x_2 - x_1$ and $y_2 = ...
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39 views

Monge-Ampere equation

I'm considering the Monge-Ampere equation in $\mathbb{R}^n$: $\mathrm{det}(D_{ij}u)=f$. I know that its linearized coefficient matrix is $\mathrm{cof}(D_{ij}u)$, i.e. the co-factor matrix of the ...
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261 views

Calculate full QR decomposition using modified Gram-Schmidt

I'm currently using the modified Gram-Schmidt algorithm to compute the QR decomposition of a matrix A (m x n). My current problem is that I need the full decomposition Q (m x m) instead of the thin ...
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97 views

Submanifold of $\mathbb{R}^4$

In the space of $2\times 2$ matrices, find explicitly the sets of matrices with 1)a single zero eigenvalue, 2) a pair of pure imaginary eigenvalues. Show that each set is a submanifold of ...
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What is the derivative of this function $\mathbb{R}\to L(\mathcal{P_n})$ at 0?

Let $\mathcal{P_n}$ be vector space of all polynomials $\mathbb{R}\to\mathbb{R}$ of degree $\leq n$. A proposition in front of me claims that, if $\pi:\mathbb{R}\to L(\mathcal{P_n})$ is defined by ...
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203 views

The norm of a diagonalizable matrix is its largest eigenvalue?

In relation to the Euclidean norm... What are the conditions for when this occurs? Is it only real symmetric matrices? When is this not the case?
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35 views

Prove or Disprove: For $V_1=V\cap V_0,$ show that $P_{V_1}=P_VP_{V_0}.$

Let $V_0$ and $V$ be subspaces of $\Omega$. For $V_1=V\cap V_0,$ show that $P_{V_1}=P_VP_{V_0}.$ I know that the above statement is not true but can't think of a counterexample. Also, if $V\perp ...
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123 views

How does adding extra row and column of ones affect a matrix's inverse?

I'm working on a homework problem, and I'm stuck. I guess my linear algebra still needs some work... I've arrived at $\mathbf{D}= \left[ \begin{matrix} \mathbf{C} & \mathbf{1}^T \\ \mathbf{1} ...
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61 views

Finding a basis for a set of points in plane and a lattice.

Suppose my domain is $\mathbb{R}^3$ where I have: a set of points $A = \left\lbrace x_i \right\rbrace_{i=1}^{n_1} $ and $n_1 > 3$. $A$ is contained in a plane (2-dimensional) $P_1$. i.e. $A ...
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508 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
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93 views

inversion of symmetrized rank 1 matrix

Consider a rank-1 matrix composed by the outer product of two vectors: xy^T. Then make a symmetric one from it: reflect the right upper part onto the lower left one. I am interested in inverting it. ...
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60 views

Using duality to establish a relationship between in two-stage linear programming

I'm currently working on a problem that involves a two-stage linear program (LP). For simplicity, I refer to the LP in first stage as LP$_1$, and the LP in the second stage as LP$_2$. The relationship ...
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101 views

Matrix subsets dimension

I'm currently studying by the book "Theory of Lie groups", C. Chevalley. On page 6, paragraph before Proposition 6, he says: "The sets $M^S$, $M^{sh}$, $M^R$, $M^S$$\cap$$M^{sh}$, $M^R$$\cap$$M^S$, ...
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Modular group representation

Does anyone know how to describe Möbius transformations with integer coefficients defined on the upper half plane in terms of $z+1$ and $1/z$? Some people call it the modular group. I would ...
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118 views

Eigenvalue decomposition under a different inner-product for orthonormalization

I would like to do an eigenvalue decomposition on a matrix $A^{\top} A$ - positive definite. Eigenvalue decomposition algorithms typically give eigenvectors which are orthogonal to each other, even ...
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68 views

Centralizers in Matrix Rings over Residue Classes

Let $p$ be a prime number and $M_2(\mathbb Z_p^t)$ the set of matrices of order $2$ over $\mathbb Z_p^t$. Let $A\in M_2(\mathbb Z_p^t)$ and $C(A)=\{X\in M_2(\mathbb Z_p^t): AX=XA\}$. What conditions ...
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204 views

Is there a great book on eigenvalues?

I keep encountering ostensibly very different branches of mathematics, only to have eigenvalues show up in each one. Is there a single book out there that presents a deep, unified account of the ...
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88 views

Find the general matrix commuting with a Jordan canonical Matrix

I need to find the most general matrix X commuting with $ J= D_{g} [J_{2}(2), J_{1}(2), J_{2}(3), J_{1}(3)] $ I also need to find the dimension of $ C(J) $ the centralizer of $ J $. I have found ...
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326 views

Commuting Matrices over Commutative Rings

Let $R$ be a commutative ring with identity and $M_n(R)$ the set of $n$ by $n$ matrices over $R$. Let $C_A(X)$ be the characteristic polynomial of $A$. Denote by $N_A$ the set (ideal) $$N_A=\{p(X)\in ...
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264 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...