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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Perspective correction from 3 points and foreshortening factor

I'm working on creating a homography 3x3 matrix to do a perspective correction of a photograph 2D piece of paper. The paper contains 3 markers (like the 3 corner markers of a QR code) in its corners, ...
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63 views

How to find $1^k+2^k+…+n^k$?

I know how to come up with $F(n, k) = \sum \limits_{p=1}^n p^k$ recursively knowing $F(n, k-1)$. But what if I want to find it in a very short time? I know how to find fast $f_n = \sum\limits_{i=1}^{...
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What are the different ways of performing Triangular matrix-vector multiplication?

Suppose we have $$\left[\begin{array}{cccc} x_1 & 0 & 0 & 0 \\ x_2 & x_1 & 0 & 0 \\ x_3 & x_2 & x_1 & 0 \\ x_4 & x_3 & x_2 & x_1 \end{array}\right] \...
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43 views

Extending the trace inner product to all matrix (real) inner products

In ${\bf R}^{n\times p}$ we have the trace inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$ which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. All inner ...
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Dimension and Basis of the $S_2$ set of symmetric matrices with $tr(A)=0, \forall A \in S_2$

For the following problem: Let $S_2$ be the set of symmetric matrices (with real entries) and zero trace. Prove that $S_2$ is a subspace of the space of all $M_{2\times2}$ matrices. ...
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Number of Solutions to a Linear Equation Mod N

Is there a formula for the number of solutions to $$a_1x_1+\dots+a_nx_n \equiv 0 \mod{N}$$ such that $(x_i,N)=1$ in terms of the coefficients $a_1,\dots,a_n$? Clearly, by Chinese Remainder Theorem, ...
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How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha \log(A)...
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70 views

Some questions about S.Roman, “Advanced Linear Algebra”

Question for those who have studied Roman's book "Advanced Linear Algebra". How self-contained is this book. Can I study determinants directly from this in context of exterior algebra and tensor ...
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49 views

Invariants of a real symmetric matrix

Problem. I have a real symmetric $n \times n$ matrix $A$ and would like to compute a set of real numbers $f(A) = (x_1, \ldots, x_m) \in \mathbb R^m$ which are invariant under multiplication of $A$ ...
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Find the codimension of $\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $\ell_2$.

Find the codimension of $A=\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $l_2$ where $S$ is the shifting operator to the right: $Se_i=e_{i+1}$. I don't quote understand ...
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19 views

Cracking any linear congruential generator

I have a linear congruential generator $X_{n+1} = (aX_n + b) \bmod 2^k $with given arguments and number $Y$. The problem is to find the smallest $i$ that $X_i = Y$ or tell that there is no such $i$. ...
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27 views

Is the perpendicular of a vector on a plane the same as the projection of a vector onto the normal?

Let x be a vector in R3 , S be a plane in R3, n be the normal vector to S. I know that the projection of x onto S is the perpendicular of x onto n. So visually, I'd imagine that the perpendicular of ...
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Are ordinary least squares coefficients 'linear' in the following sense.

If I have two sets of noisy data, with same number of points in each set and measured at the same set of x positions, then carry out a polynomial least squares fits on each set of data, are the ...
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48 views

Relationship between row space and orthogonal component of kernel of complex vector space

When we consider the real vector space, row space is equal to orthogonal complement of the null space (kernel). This fact can be proved as follows. Let's consider linear map $A : \mathbb{R}^n \...
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40 views

Explicit example of Gershgorin circle theorem edge case

The Gershgorin Circle Theorem states that if the union of $m$ of the discs is connected, and disjoint from any discs not in the union, then it contains $m$ eigenvalues of the matrix. I am looking for ...
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Calculate the matrix of a linear opertor that transforms a vector to a Hankel matrix

I would like to calculate the matrix associated to a linear operator $\mathbf{R}$ that transforms a vector $\mathbf{x}\in\mathbb{R}^N$ into a Hankel matrix $\mathbf{H}\in\mathbb{R}^{N-Q+1\times Q}$ ...
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39 views

Matrix comparison depend on one scalar variable

Let $A$, $B$, $K$ be $n\times n$, $n\times m$ and $m\times n$ matrix respectively. $\alpha_i>0$ is a scalar. Consider the following matrix: $H_i=\int_0^\infty e^{(A+\alpha_i BK)^Tt}(\alpha_iI+\...
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53 views

Fastest way of find roots of polynomial defined over a finite field

Suppose we have polynomial $G(x)$ of degree $d$, where $d$ is a large value (e.g. $10^6$). The polynomial is defined over a finite field $\mathbb{F}_p$ for a large prime number $p$ (e.g. $p$ is 200-...
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How many solutions does a consistent linear system of three equations and four unknowns have? Why?

$4$ vectors in $\mathbb R^3$ yield $3$ equations in $4$ variables. At most only $3$ of the vectors will be independent in $\mathbb R^3$. So $4$ vectors in $\mathbb R^3$ span $\mathbb R^3$ if $3$ of ...
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21 views

finding orthogonal complement in the given cases

let $V = \Bbb R_5[x] $ with the inner product defined by $\langle p(x),q(x)\rangle =\int_{-1}^1 p(x)q(x) \,dx$. find $W_\bot$ if (1) $W=\{p(x)\in \Bbb R_5[x] :p(x)=p(-x)\}$ (2) $W=\{p(x)\in \Bbb ...
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Distance to subspace spanned by two vectors

i am finding difficulty in the following problem: let W=span{$\frac{1}{\sqrt 2}$(0,0,1,1),$\frac{1}{\sqrt 2}$(1,-1,0,0)} be a subspace of the euclidean space $\Bbb R^4$. then,what is the square of ...
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34 views

Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually I'm not exactly looking into bipartite but the ...
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77 views

Prove that the differentiation operator on the real vector space of polynomials of degree at most $n$ is nilpotent.

Let $P$ be the real vector space of polynomials $p(x)=a_0+a_1x+\cdots+a_nx^n$ of degree at most $n$, and let $D$ denote the derivative $\frac{d}{dx}$, considered as a linear operator on $P$. $\...
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25 views

Intersection of halfspaces and hyperplanes

If $H_1$, $H_2$ and $H_3$ are hyperplanes in an $n$ dimensional vector space $V$ then I want to prove that the linear span of $ H_1\cap H_2^+\cap H_3$ is $H_1\cap H_3$. Clearly the span is contained ...
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47 views

Expectation of a special form of quadratic form

Let $\mathbf x$ be a $n\times1$ random variable, $\mathbf s$ be a vector of size $3\times 1$ and $A$, $M_1$, $M_2$ and $M_3$ be $n\times n$ matrices. What is the following expectation with respect to ...
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58 views

Show that scatter matrix is possitive definite

If we have a sample with size $n$ $(\mathbf x_1,\dots, \mathbf x_n)$, where $\mathbf x_j \in \mathbb R^p$, the scatter matrix is defined as $$ S=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x}...
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Property of pairs of disjoint convex polytopes in $\mathbb{R}^n$

I am having trouble proving the following property. Maybe the result already exists, but I could not find anything on the topic. Proposition: Let $P_1$ and $P_2$ be two disjoint polytopes of $\...
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92 views

Upper bound on Frobenius norm of inverse of positive definite, symmetric matrix

Let $\Sigma$ be a symmetric, positive definite $n \times n$ matrix. I want an upper bound on the Frobenius norm of $\Sigma^{-1}$ that does not involve calculating the determinant of $\Sigma$. The ...
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44 views

Determine and classify all equilibrium points of this non-linear DE

Consider the DE $\begin{cases} \dot{x}=-2x(x-1)(2x-1)\\ \dot{y}=-2y \end{cases}$. Determine all equilibrium points and classify these. Choose between a saddle point, (in)stable nod, center or a (in)...
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Find length of sequence with fibonacci principle

Let assume we have sequence: $${l_{n + 1}} = {l_n} + {l_{n - 1}} $$ $$\begin{array}{l}{l_0} = 0\\{l_1} = 01\\{l_2} = 010\end{array}$$ Our goal to get $|{l_n}|$ (length). For ${l_0}$ it's 0, for ${...
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31 views

Proving that a map is zero

Let $A,B$ be two banach spaces and $\Omega, \nu, \Sigma$ a measure space. Consider the map $F \colon \Omega \to \mathcal L(A,B^*)$. If $F(\omega)(a)=0$ for all $a \in A$. then why does one need $A$ to ...
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Multiplication of Matrices in a Hilbert Space

So I was having a discussion with a friend as follows: Let $\mathcal H$ be a Hilbert space. Let $\mathcal H^{\otimes n} = \mathcal H \otimes \mathcal H \otimes \cdots \otimes \mathcal H$. $\mathcal H^...
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Books with similar coverage to Linear Algebra Done Wrong

Axler's book is great, but for my immediate purposes, it isn't suitable. I've been looking at the Table of Contents of Linear Algebra Done Wrong by Treil starting at p. 5 of this document but there's ...
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If $0 \in W(A)\mathop \Rightarrow \limits^? W(A) = \left\{ {{x^*}Ax:x \in {C^n},{x^*}x \le 1} \right\}$

Let $A\in M_n$ and $W(A)$ be numerical range of $A$. Suppose $0\in W(A)$. Why does $W(A) = \left\{ {{x^*}Ax:x \in {C^n},{x^*}x \le 1} \right\}$?
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How do we deduce that $c_1=c_2=0$?

Theorem The boundary value problem $$y''+p_1(x) y'+ p_2(x)=f(x)(1)\\ a_1 y(x_0)+a_2 y'(x_0)= b_1 y(x_1)+b_2 y'(x_1)=0 (2)$$ has a unique solution iff the corresponding homogeneous problem ( $f \...
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46 views

Trace inequality $tr(|XY|) \leq \|X\| tr(|Y|) $

Why does one have $tr(|XY|) \leq \|X\| tr(|Y|) $ for any complex matrices? I do know that Cauchy Schwarz establishes $|tr(X^*Y)|\leq \|X\| \|Y\|$. Ok so far I have $\langle u,X^*Xu \rangle=\langle Xu,...
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How to do a Hermitian matrix decomposition?

I want to recover the matrix $A$ out of the Hermitian matrix $H$ such that: $$H = A\cdot A'$$ where : $H$ is a hermitian matrix. $A$ is a non-square matrix. $A'$ is the conjugate transpose of the ...
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Linear programming of integers

Suppose that we have a linear programming problems involving two integer variables. Many textbooks state that the optimal solution will occur at the corner point. However, what if the corner point ...
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Subalgebras of matrices property

Let $g$ be a Lie subalgebra of $gl_n(\mathbb{C})$ which has the propety that if $a\in g$ then also $a^\dagger\in g$ (where $a^\dagger$ is conjugate transpose). I want to show that if $a$ is an ideal ...
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The variance of a sum of random vectors

There are $n$ vectors each containing exactly $q$ random variables as elements. Each vector is denoted I$_k$. Each variable within the vector has its own (normal) probability distribution, and the ...
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Reference for the “geometry” or “arrangements” of subspaces of a vector space?

Inspired by Section $5$ of Chapter $1$ in Kostrikin & Manin's famous "Linear Algebra and Geometry", I am searching for a book or paper on the geometry or arrangement of subspaces in a finite-...
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39 views

Find the number of combinations for binary vectors in the following case

Let $\mathbf{v}$ be an $N$-dimensional binary vector. The first $m$ coordinates of $\mathbf{v}$ belong to a category, termed as "category a", and the last $n$ ($=N-m$) coordinates belong to another ...
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30 views

Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} &...
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If $A^2$ is diagonalizable then A is also…

I tried to dis/prove the following: A is a 2x2 complex matrix 1) If $A^2$ is diag. then $A$ is also. I mostly tried using $P^{-1} A^2 P = D$ and found out that: $D=P^{-1} A PP^{-1} A P = \sqrt{D}\...
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31 views

Is $\operatorname{End}(V)\simeq M_n(D)$?

Let $D$ be a division ring, and $V$ a vector space over $D$, such that $\dim_DV=n$ where $2\le n<+\infty$. Then I must prove that $R:=\operatorname{End}(V)$ is a prime ring (a ring $R$ is prime if ...
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34 views

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?

Does there exist a self-adjoint operator whose spectrum is just the continuous spectrum?(i.e. no point spectrum and no residue spectrum) If not, please prove it.
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calculate triangular form of a matrix

The following I don't understand. I would really appreciate any help! I want to find a flag basis for the following matrix: $A=\begin{bmatrix} 3&0&-2\\ -2&0&1\\ 2&1&0 \end{...
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42 views

Are linear combinations of random vectors linear independent?

If I have $n$ random vectors $x_1,\ldots,x_n\in\mathbb{R}^n$, they are linear independent with high probability. What can be said about linear combinations of these vectors? I.e. does for $n$ ...
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39 views

Expected number of binary random vectors for dimension reduction

Assume that Trent has a binary random vector generator which creates vectors of length $n$. Each element of these vectors can be either zero or one with equal probability. Trent creates a set of $M$ ...
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72 views

A general form of a matrix $A$ raised to a natural power, $A^k$.

Suppose I have the $2\times 2$ matrix: $$A = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$$ I want to find the general matrix $A^k$ where $k\in \mathbb{N}$. I.e I want to ...