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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Some questions about a “relaxed” invariant probability problem $|\mu(P-I)|\leq \epsilon$

Let's consider the set $\mathcal{M}=\{\mu:|\mu(P-I)|_1\leq \epsilon\}$ where $\mu$ is a probability vector, $P$ is the transition matrix of a discrete homogeneous Markov chain, $I$ is the identity ...
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60 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
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150 views

Matrix of Discrete fourier transform $F^4$ is identity

I already showed that Discrete fourier transform matrix is unitarian matrix. Now I would like to show that $F^4$ is identity. On wikipedia is written: "This can be seen from the inverse properties ...
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383 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
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66 views

For any linearly independent vectors $x, y$, there is such a norm that $||x||_*>||y||_*$

Today I've seen in my class that: For any linearly independent vectors $x, y$, there is such a norm that $||x||_*>||y||_*$ Our lecturer called it Benchmark theorem. I wanted to learn more ...
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147 views

Existance of inverse operators for Hermitian adjoint operators

I have one assumption that I can't prove. Maybe there are some other conditions which I didn't take into account. My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and ...
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51 views

Forming $L^2$ projection of function

I am bit unsure if I'm attacking the following problem correctly: Let $n$ be a positive integer, and let $f$ be a continuous function defined on $[0,1]$. Let $h_k (t) = \sqrt{n} \phi(nt - k)$, where ...
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334 views

Jordan Canonical Form and Minimal Polynomial

I was wondering what is the relationship between minimal polynomial and the Jordan Canonical Form. Before given a matrix, all you need is to compute the characteristic polynomial to determine the ...
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32 views

finding the decomposition of Laplacian matrix with position of zero elements unchanged

I'd like to know whether it's possible to find the decomposition of a Lapalacian matrix $A$ $B^TB = A$ where $B$ has the same dimension with $A$ and the position of zero elements in $B$ is the same ...
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91 views

Linear Programming: Modifying Coefficients of the Objective Function

Consider a final tableau with entries: Row 1: 0,(-1/2),1,1,2,0,-1 Row 2: 1,(1/2),0,2,-1,0,-2 Row 3: 0,2,0,-1,(-1/2),1,3 Basic variable values (4,2,1) and objective function coefficients ...
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Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?

I come across a theorem that $f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda ^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$ where $a_{i}$ is the sums of all ith order ...
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76 views

Need help with proof.

I have come up with an answer, but im not sure if its right if someone would please take a look i would really appreciate it. Suppose V is a real vector space, $T\in L(V, V )$, and V has a basis ...
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108 views

Linear algebra:P2 - orthonormal basis of V and $\mathbb{R^3}$

In $\mathbb{R}^3 $ with $ \underline{N}= \begin{pmatrix} 1\\ 2\\ 1 \end{pmatrix} $ and the subspace $V={\underline{N}^\bot}$. From Linear Algebra:P1 - Dim(V), linear independence ...
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93 views

Flatten kronecker product in CAS?

I am a new user of Maxima, and I need to trace the elements of a big messy kronecker product of symbolic matrices. I tried the following to get my feet wet, but I don't get a simple, flat matrix -- ...
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43 views

Show that if S = {b1,b2,…,bk} in any generating set for V, then the pivot columns of the matrix …

Let $V$ be a subspace of $\mathbb{R}^n$. According to the extension theorem, a linearly independent subset $L = \{u_1,u_2,...,u_k\}$ of $V$ is contained in a basis for $V$. Show that if $S = ...
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87 views

Finding a basis which is orthonormal with respect to symmetric bilinear form

Let $X=\{a\in M_4(\mathbb{C}): a^T=-a\}$ be the $6$-dimensional vector space of matrices. Define the Pfaffian as $$ \mathrm{Pf}(a)=a_{12}a_{34}-a_{13}a_{24}+a_{14}a_{23} $$ There is an associated ...
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72 views

Linear Dependence problem

Lets suppose we have a set of vectors $\{(1 ,0, 0, 0 ) , (0, 1, 0, 0 ), (2, 0, 0, 0 )\}$. By definition this set is linearly dependent because we can find constants $c_1,c_2, c_3$ ( such that all ...
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132 views

What is the name for a non-square permutation matrix?

Consider a matrix that selects and permutes some but not all of the entries of a vector. That is a binary $n\times m$ matrix, where $n<m$, with a single one per row, for example ...
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55 views

how to transform a space to optimally separate data

I'm not a mathematician, but I need mathematical help. Apologies in advance for notational anomolies and/or poor tagging :-) I have a classification problem where data points $x \in \mathbb{R}^n$ ...
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64 views

Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to ...
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207 views

Criterion for detecting rank-deficiency via QR decomposition?

I apologize in advance if this is an ill-posed question -- I'd appreciate advice on what pieces are missing as much as an answer. I'm solving a system like $P \approx X Y^T$, where P is a large ...
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160 views

volume of linear transformations of Jordan domain

Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear transformation and $R\in \mathbb{R}^n$ be a rectangle. Prove: (1) Let $e_1,\ldots,e_n$ be the standard basis vectors of $\mathbb{R}^n$ (i.e. ...
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68 views

'Injectivity' of a bilinear map restricted to the set it can generate starting from a given vector

Here is a problem I'm stuck with: Let $V$ be a vector space (on a field $F$) of finite dimension $n$, $v\in V$ and $\mu : V\times V \mapsto V$ a bilinear application determined by its action on the ...
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32 views

Inverse of $(U^H X U + D)$ where U is unitary, X and D diagonal

Given complex unitary matrix U, and full rank diagonal matrices X and D with positive entries. I'm looking for an efficient way to compute: $(U^HXU+D)^{-1}$ The matrix inversion identity doesn't ...
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354 views

Collinearity, coplanarity and determinant

I don't get what the question is asking me. I'm confused why they add '1's to the matrix. Anyway here's my attempt. For part (i), my analysis is that since P2, P3, P4 are not collinear, therefore ...
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37 views

Let E a projection,calculate $f (E)$

Let $V$ a vector space over the field $F$ and E be a projection of $V$ and f is an element of $F[t]$, of grade greater than or equal to 1, calculate f (E) and determine the relation between ...
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84 views

Why does the map $x^2$ have constant rank?

I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
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97 views

The value interpretation of eigenvectors.

My question is may be strange but I wanna lie it any way. The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...
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49 views

When can a matrix with negative entries have a completely non-negative dominant eigenvector?

Perron-Forbenius obviously answers this question for positive and for certain non-negative matrices. I want to know whether these conditions can be weakened at all. In other words, what, if anything, ...
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45 views

Relation between the coefficients in the different basis.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ we can represent this polynomial $f$ in two basis, Monomial basis and hermite basis. How can we get the relationship between coefficients of $f$ in both ...
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213 views

Transforming matrices in a differential equation

This is from Dupont et al., "Simplified density-matrix model applied to three-well terahertz quantum cascade lasers", PRB 81, 205311 (2010): Equation (3) (...) can be rewritten as a 16x16 system of ...
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56 views

Issue with tridiagonal matrix factorization

So let's assume I have an arbitrary mxm tridiagonal matrix made up of real numbers. How many flops are needed to get its QR factorization (assuming I'm using the householder method?) How would one go ...
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Can we express a SPD matrix $S$ in terms of $S^{2}$ in a different manner to solve a convex problem?

I have to find the Symmetric Positive Definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ which has been proven to be convex in the ...
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188 views

How is the Quaternion multiplication derived?

Quaternion multiplication seems suspiciously similar to the cross product. How is it derived? Here is a description of the multiplication: Let $Q_1$ and $Q_2$ be two quaternions, which are defined, ...
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169 views

How to determine all vectors $b$ for which $Ax = b$ has a solution? Do the columns of $A$ span $\mathbb{R}^3$?

Let $A = \left(\begin{array}{rrr|r} 1&1&-15&36\\ 1&2&-10&41\\ 1&2&-9&42 \end{array}\right)$. Here is the row reduction: $A = \left(\begin{array}{rrr|r} ...
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55 views

Embedding an $n$-simplex in $\mathbb{Z}^n$.

I am trying to understand the proof of embedding an $n$-simplex in $\mathbb{Z}^n$ for specific values of $n$. The proof can be found here. I am stuck on what is meant by "the reflection with axis ...
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46 views

Isomorphisms betweenVerma modules over a semisimple Lie algebra

Fix a finite dimensional, semisimple Lie algebra $L$ and denote the Verma $L$-modules by $V(\lambda ')$ where $\lambda '$ are corresponding weights. Assume that there is an isomorphism between two ...
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61 views

Curvature of surface

So lets say I have a mesh and for each face I have the position of its $3$ vertices and the area of the face. So let's say I have a point $p$ on this face and a vector $v$ that goes from $p$ to the ...
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191 views

Vector Projection and Cross Product

Let there be $w, u, $ and $v$, such that: $$w \times u = \langle1, 3, 5\rangle$$ $$w \times v = \langle 2, 4, 6\rangle$$ Find: $$v \cdot (((u \times w) \times v) + \text{VP}uv(w)) + ((u ...
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40 views

pseudoinverse under change of norm

Let $X$ be a Hilbert space. Let $T : X \rightarrow X$ be a linear mapping. Suppose we have two scalar products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ on $X$. Let $T_1$ and ...
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60 views

Optimizing over norms of set of equations.

I have a set of N real-valued finite-dimensional vectors $\mathbf{v}_i$ and target norms $y_i$ and I am trying to find a linear transformation matrix $L$ such that the norm of the transformed vectors ...
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238 views

Characteristic polynomials of linear transformations/matrices.

I just have a quick question about characteristic polynomials. It is defined as $$\chi(t) = \det(A - tI)$$ for a matrix $A$. Similarly for a linear transformation using its matrix with respect to some ...
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45 views

Characterizing (or deriving) the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$, $$f(x,y) = e^{\imath\pi x g(y)}$$ where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$ ...
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116 views

Showing that a non-standard basis for Polynomials is a basis

Let $B_1 = \{ 1 + x, 1 - x \}$ be a subset of $P_1$ and $B_2 = \{ 1 + 2x, 2 + x\}$ be a subset of $P_1$. Both $B_1$ and $B_2$ are bases for $P_1$ , where the usual left to right ordering is assumed. ...
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161 views

Multi-dimensional array decomposition

My question is about decomposing a muti-dimensional array into a product of matricies. To ask the question I will work towards the tensor, and then ask the question about the reverse process. Let ...
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112 views

3 nonzero distinct eigenvalues, part 2

This is an attempt to generalize the answer to a previous question Consider the $n \times n$ matrix $$A=\left[ \begin{array}{cccc} 0 & \frac{1}{n-1} & ... & \frac{1}{n-1} \\ 1 & 0 ...
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83 views

necessary and sufficient conditions for a subset to be the graph of a linear operator

Let $X$ and $Y$ be two linear vector spaces. Find necessary and sufficient conditions for a subset $G\subset X\times Y$ to be the graph of a linear operator from $X$ into $Y$. The definition for the ...
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41 views

orthogonal base in unimodular lattice

Let $\Lambda$ be an unimodular lattice with a quadratic form $(-,-)$ of signature $(m,n)$ , $m,n>0$. I know that, fixed a base $e_1,\cdots,e_{m+n}$ for $\Lambda$, the matrix which has entries ...
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251 views

Weighted linear least squares parameter covariance

I am currently trying to figure out the parameter covariance for a weighted linear least squares problem where $$y = X\beta$$ The parameters for which my objective function is lowest are given by ...
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42 views

How do we prove that the left singular vectors form a basis for the column space of $A$?

I need to prove that left singular vectors form a basis for the column space of $A$ and how do we find a basis for the null space of $A$ using singular vectors. Please help!