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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Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
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146 views

How does pointwise multiplication of two matrices affect their eigenvectors?

More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$. What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ ...
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158 views

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
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60 views

Seeking diagonal $D$ such that $AD^2B$ is positive definite (A and B are symmetric and p.d.)

Let A,B be two real symmetric, non-commuting positive definite matrices. Then the product AB is not positive definite (in the sense that ${\bf x}^TAB{\bf x}$ may be anything). I'm trying to find ...
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311 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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90 views

Convexity of $S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ^{n}\right\}$

i've to prove that the following set is convex: Let $A \in \mathbb{C}^{n \times n}$ $$S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ...
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224 views

Alternative introduction to tensor products of vector spaces

One of the main obstacles in understanding the tensor product is that, unlike many other algebraic structures, you cannot really get hold of its element structure. This confuses many beginners. The ...
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78 views

finding the algebraic dimension of $\ell^p$ spaces

I want to know "how we can find the algebraic dimension(the cardinal number of the Hamel basis) for $\ell^p$ spaces." What can we say about $\ell^p(I)$, where $I$ is an infinite set?\ Moreover, for ...
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36 views

Reference for Generalized Eigenvectors

I am looking for references on generalized eigenvectors and Jordan matrix representation. I would like a brief but complete introduction of this concepts with a nice treatment of the most important ...
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99 views

Consistency of different ways to define the tensor product

It seems my trouble with understanding tensors stems from the following statement: More specifically, the statement: Namely, given $B: V \times W \to U$ and $\xi: U \to \mathbb{R}$, $\xi ...
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70 views

co and contravariant vectors, their difference and properties

Very often when talking about covectors, co- and contravariant stuff, it's mentioned that there is no difference in "normal" linear algebra. That the difference only comes "when dealing with curved ...
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222 views

Linear Algebra, Eigenvalues and Eigenvectors Exercise

I have a question from an exercise. I am given a vector space over the field $\mathbb{R}^{3}$ with 2 dimensions and I am asked to find a basis of eigenvectors. I found the eigenvalues but I have ...
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74 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
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58 views

Blocking set for cosets of codimension $2$

In this paper following theorem is proved: If $V$ is vector space of dimension $n$ over a finite field $F$ of $q$ elements then any subset of $V$ which meets every hyperplane of $V$ contains at least ...
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83 views

Does this characterize the operator norm of the inverse?

Let $A$ be an invertible operator (bounded with bounded inverse). Then $$\frac{1}{\|A^{-1}\|} = \inf\left\{\frac{\|Av\|}{\|v\|} : v \neq 0\right\}$$ I believe I have a proof as follows, but I just ...
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47 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
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116 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
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147 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
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49 views

Are these vectors a base of the Subspace?

$a=(1,2,3,0)$ $b=(0,3,2,4)$ $a$ and $b$ form the subspace $U$ is the base $B(a,b)$ a base of $U$? My guess is yes, because they are linearly indpendent?
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Does the cross section of $[-1,1]^n$ on a $k$-dimensional subspace always contains a rotated image of $[-1,1]^k$?

This question is inspired by a recent bounty question, but the two questions are different and solving this one, I believe, will not lead to an answer of that bounty question. Suppose $n>k\ge1$ ...
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130 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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54 views

Particular determinant made of powers of algebraic numbers is nonzero?

Let $P$ be a degree-two polynomial, with roots $\alpha,\beta$. Is there a simple condition on $P$ (or on $\alpha,\beta$), equivalent to the following : $$ ...
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75 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
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68 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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134 views

Understanding a sentence in a paper.

I am trying to read this paper. At page $34$, the authors define a symplectic form $\omega$ on some $\mathbb{C}$-vector space $V\oplus W^*$ (I don't want to go too much into the details here, because ...
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277 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
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104 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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83 views

Identifying the joint distribution from some values of $t \cdot X$

Suppose that $S$ is a subset of $\mathbb{R}^n$ and $X, Y$ are $\mathbb{R}^n$ valued RVs. We already know that $X$ and $Y$ are equidistributed iff $t \cdot X=^d t\cdot Y$ for all $t \in \mathbb{R}^n$. ...
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86 views

Smallest value that a certain variable can take in a system of equations.

Consider the solutions $(x,y,z,u)$ of the system of equations: $$\begin{cases} x+y=3(z+u)\\ x+z=4(y+u)\\ x+u=5(y+z)\\ \end{cases}$$ where $x,y,z \text{ and } u$ are positive integers. What ...
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100 views

Linear dimension of banach spaces

Let $X$ be some vector space (over $\mathbb{C}$). Note that if $X$ is of finite dimension we can identify $X$ with $\mathbb{C}^n$ for some natural $n$ and endow it with a norm $||x||=|x_1|+...+|x_n|$. ...
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87 views

Show $r(F)=r(F^2)$ implies $Im(F) \cap Ker(F)=\{0\}$

I wonder if I've made some mistakes in the proof of the following or if there is some simpler solution. Problem: Let $V$ be a finite dimensional vectorspace and $F:V \rightarrow V$ a linear operator. ...
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Which polynomials are characteristic polynomials of a symmetric matrix?

Let $f(x)$ be a polynomial of degree $n$ with coefficients in $\mathbb{Q}$. There are well-known ways to construct a $n \times n$ matrix $A$ with entries in $\mathbb{Q}$ whose characteristic ...
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219 views

The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
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94 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
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64 views

Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$ F:S\times T\longrightarrow K $$ where $K$ is a topological field ...
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1k views

Proof of rank nullity theorem

I read about rank nullity theorem (with proof) but then tried to prove it in different way. Please can you read my proof and tell me if it is correct? The rank nullity theorem: If $T:V\to W$ is a ...
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540 views

Upper bound for smallest eigenvalue

I am looking for a (simple) upper bound for the smallest eigenvalue of an $n\times n$ matrix, involving determinant or trace or something else that can be easily computed. I've got an upper bound from ...
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66 views

Subgroup consisting of unipotent elements centralizes a flag

Is there a reasonably elementary and short proof that a subgroup of consisting of unipotent matrices over a field centralizes a flag? By elementary, I mean accessible to students who have had a ...
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97 views

Structure of a fuzzy subspace

Let $V$ be a vector space over a field $F$ and let $f$ be a function from $V$ to the interval $I:=[0,1]$ satisfying the condition that for any $a \in I$ the set $V_a:=\{v \in V | f(v) \ge a\}$ is a ...
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139 views

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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98 views

Linear independence of $\cos(n\theta)$

I was trying to see if the cosines of the (certain) integer multiples of a certain angle were linearly independent over $\mathbf{Q}$. In particular I was looking at when $\theta = ...
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444 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
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255 views

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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40 views

what to do if it's not direct sum?

Suppose $X=Y+Z$ is Banach, $Y$ and $Z$ are closed subspaces. I want to show there exists $\alpha>0$ such that $\forall x \in X, \exists$ $y \in Y$ and $z \in Z$ such that $x=y+z$ and $\|y\|+\|z\| ...
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96 views

Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
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169 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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339 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
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274 views

Low-rank approximation to the Graph Laplacian matrix of a regular grid.

As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
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Is sub-vector an established mathematical entity?

Reading the following paragraph I was wondering how this entity that the authors [1] call a sub-vector should be named. In Matlab a sub-vector has to be contiguous. Let ...
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189 views

Understanding Blackwell's Approachability Theorem

I'm not super solid on my linear algebra, so I am getting lost in the discussions of halfspaces. Can someone give me an intuitive explanation (possibly with a concrete toy problem) of Blackwell's ...