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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Closed Operations in Spherical Space

Euclidean vector arithmetic (addition/subtraction) is not closed under a spherical space. For example: $\mathbb{S}^2=\{v \in{\mathbb{R}^3}|\|v\|=1\}$ We have $(1, 0, 0)\in\mathbb{S}^2$ and $(0, ...
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182 views

Find the signature of the quadratic form

Very simple question but something doesn't make sense to me. We are given a quadratic form (bilinear map but on the same vector twice): $Q(v) = v^t *\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 ...
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131 views

Matrix equation $AX=XB$

For $A,B \in \big( \mathrm{Mat}_{n}(\mathbb{C}) \big)^2$, I know that there exists $Y \in \mathrm{Mat}_{n}(\mathbb{C})$, $Y \neq 0$, such as $AY=YB$ if and only if $\mathrm{Sp}_{\mathbb{C}}(A) \cap ...
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2k views

How to find the vector perpendicular to a line and passes through a point

How to find the vector perpendicular to a line that passes through a point that does not lie on that line?
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39 views

let $B(,)$ and $B'(,)$ be inner products over $\mathbb R$. show there is $c \in \mathbb R$ s.t $B(u,u) \leq cB'(u,u)$

As the title says, let $B(,)$ and $B'(,)$ be inner products over $\mathbb R^n$. show there is $c \in \mathbb R^n$ s.t $B(u,u) \leq cB'(u,u)$ for all $u \in \mathbb R^n$. I have been thinking about ...
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39 views

how to get theta for a product of two orthogonal matrices

Let $O(2)$ be the Lie group consists of all $2 \times 2$ orthogonal matrices, i.e. all matrices such that their transpose is equal to their inverse. The operation is the usual product of matrices. It ...
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49 views

Computing the Permanent of a Matrix in Terms of Permanents of its Sub-matrices

Say I have a matrix \begin{align*} G &= \left( { \begin{array}{cc} G^{\prime} & \vec{u} \\ \vec{v}^T & d \end{array} } \right) \end{align*} where $\vec{v}^T$ and $\vec{u}$ are row and ...
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62 views

GARCH Covariance Matrix in Matlab

I need to have the GARCH covariance matrix for my project and I want to know if this formula is correct: in matlab using garchfit , we have "Innovations" and "Sigmas".To obtain a GARCH(1,1) ...
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165 views

Voronoi-Diagram - split concave polygon

I have a concave polygon c and a set of points p[]. What I need is to have a set of polygons splitted by the voronoi-diagram. My problem is the following: I already found a library to calculate the ...
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91 views

An inequality about rank of matrices

Suppose $A$ and $B$ are $m \times n$ matrices, prove that: $$\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(A+B) \ge \operatorname{rank}\pmatrix{A & ...
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462 views

What does this linear algebra notation mean?

I'm trying to prove that a particular $V$ is a $\Bbb{Q}$-vector space. The question says to take the element $0_V = 1$, the function $+_V : V \times V \to V$ given by the function $[x +_V y = xy]$, ...
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115 views

Eigenvalues of the Google Matrix

Let $c \in \mathbb{C}$ and $x, v \in \mathbb{C}^n$ satisfy $v^*x= 1$ (where the start means conjugate transpose). Given that a square ($n\times n$) matrix $A$ with complex entries has eigenvalues ...
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91 views

Is the set of all matrices in M(n;R) all of whose eigenvalues satisfy the condition |λ|≤2. compact?

Is the set of all matrices in $M(n; R)$ all of whose eigenvalues satisfy the condition $|λ| ≤ 2.$ compact?
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101 views

Determine whether or not the set of real numbers $\mathbb R$, together with the operation* defined by $a ∗ b = (a+b)/3$, forms a group. [closed]

How would one show this is associative? I think I've shown it is closed but I'm not sure how to show it is associative.
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123 views

Calculate the determinant of given matrix

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
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61 views

find invariant subspace of polynomials

$(L(t)f)(x)=f(x-t)$ I know that $L$ is representation of the group $\mathbb{R}$ in space continuous functions defined on the real line. Find all the invariant subspaces of polynomials of $L$. ...
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47 views

Sparse LU Decomposition

So there are several ways to carry out sparse LU decomposition on a computer, such as the Block LU decomposition algorithm, left/right looking algorithm , etc. My question is how do these methods ...
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105 views

Solving the General Equilibrium with $4$ equations and $4$ unknowns

I have to solve four equations to solve the equlilibrium prices for the two countries: $\frac{2p_1}{w_1} + \frac{p_1}{w_2}= \frac{48w_1^2 + 4p_1^2+p_2^2}{8p_1w_1}+ ...
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43 views

Is the following a reflection matrix

Suppose that $A,B \in \mathbb{R}^{n \times n}$. Let $A^2=A$ so that $A$ is a projection, $B^2=I$, and $B=2A-I$. Is it true that $B$ represents a reflection?
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190 views

Decompose $A=D+N$ with $DN=ND$, $N$ nilpotent, $D$ diagonalizable

Can anyone help me out with the following question: For the matrix $A$ give a diagonalizable matrix $D$ and a nilpotent matrix $N$ so that $A=D+N$ and $ND=DN$. $\begin{bmatrix} 1 & 4 \\ -1 & ...
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2answers
109 views

How do i prove that every linear operator between finite-dimensional Hilbert spaces is bounded?

When I learned basic linear-algebra, "adjoint" was only defined for linear operator between finite-dimensional inner product spaces. Right now, I'm studying Hilbert spaces and I want the past ...
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1answer
81 views

Finding non-negative matrices, 0 on the main diagonal for which this positive vector is invariant.

This is a sort of reverse eigenvector problem. Usually, given a matrix, we want to describe its eigenvalues. Here -- given a vector, we'd like to determine matrices (satisfying some conditions) for ...
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59 views

Find two linear homogenous equations that are equivalent to the span of a line in 3 dimensions.

I'm working on an exercise in linear algebra and I'm stumped by part of it. Say I have a line in 3 dimensions that can be represented as $\operatorname{Span}\lbrace[1,-2,-2]\rbrace$. The book says ...
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477 views

Direct summand of skew-symmetric and symmetric matrices

Let $W_1$ be the subspace of $\mathcal{M}_{n \times n}$ that consists of all $n \times n$ skew-symmetric matrices with entries from $\mathbb{F}$, and let $W_2$ be the subspace of $\mathcal{M}_{n ...
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67 views

Characterization of linear functions in $\mathbb{R}$ using distance

First of all, by a linear function in $\mathbb{R}$, I mean a function $f:\mathbb{R}\rightarrow\mathbb{R}$ of the form $f(x)=ax+b\ \forall x\in\mathbb{R}$, where $a,b\in\mathbb{R}$ (not in the linear ...
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1answer
606 views

Product of two symmetric matrices with eigenvalues all 0 or 1 is idempotent

Assume $A$, $V$ are symmetric and $V$ is positive definite. If $AV$ has eigenvalues that are all zero or one, show $AV$ is idempotent. My proof so far (haven't used the symmetric property or the ...
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1answer
83 views

A lower bound on the second largest eigenvalue of a $4\times 4$ matrix.

I have a $4\times4$ non-symmetric symbolic matrix with the following characteristics: there are positive, negative, and zero entries; the determinant and trace are both positive; under relevant ...
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1answer
36 views

Showing that the following functions aren't inner products

Show that the following functions aren't inner products: $$1. \ V=\mathbb R_5[x];\ \ \langle p,q\rangle =\sum^5_{i=0}p(q(i)) \\2. \ V=\mathcal M_{2\times2}(\mathbb R);\ \ \langle ...
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1answer
42 views

How do I calculate the intersection between two cosine functions?

$f(x) = A_1 \cdot \cos\left(B_1 \cdot (x + C_1)\right) + D_1$ $g(x) = A_2 \cdot \cos\left(B_2 \cdot (x + C_2)\right) + D_2$ Is it possible at all to solve this analytically? I can start ...
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85 views

Trace zero of a product of non square matrices

Question: Prove that if $S\in M_{n*m}\Bbb F, T\in M_{m*n}\Bbb F$ and $tr(ST)=0 \Rightarrow S=0,T=0$ Things I did I read alot of questions here and figured that the matrices commute if their trace is ...
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190 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
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1answer
116 views

Composition of orthogonal projection

Given $\gamma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (rotation around $o$) and $\sigma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (reflection in one of the lines through the origin), I have to show that ...
2
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1answer
119 views

Jordan normal form - Real matrices

Given the matrix $A = \begin{pmatrix} 7 &1 &2 &2 \\ 1 &4 &-1 &-1 \\ -2 &1 &5 &-1 \\ 1 &1 &2 &8 \end{pmatrix}$. I found its characteristic polynomium, ...
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1answer
28 views

Distribution of multivariate Gaussian conditional on value of linear function

Given a Gaussian random vector $X \in \mathbf{R}^p \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma)$, a matrix $\mathbf{K} \in \mathbf{R}^{q \times p}$, and a vector $y \in \mathbf{R}^q$, I'd like ...
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833 views

Is horizontal line always parallel to the x-axis?

I am having an argument about the answer of the following question with my friend: Determine if the following sentence is always, sometimes or never true: - A horizontal line is parallel to the ...
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2answers
647 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...
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1answer
84 views

What is DifferenceRoot?

In a question I asked earlier today in an effort to simplify the equation on the left below, I was pointed toward Wolfram Alpha. I got this back: I honestly have no idea what the right half of ...
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2answers
143 views

Calculating intersection between a linear function and a cosine function

I'm trying to calculate the intersection between the two following functions: $y = kx + m$, $y = A \cos(B(x+C)) + D$. To find the intersection I start by assuming that both of the ...
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255 views

What is symmetric square root of matrix?

In one publication I found a following part: $\textbf R^{-1/2}$ is a symmetric square root of matrix $\textbf R^{-1}$ I know what is a square root of matrix, but what exactly is symmetric ...
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37 views

Is $Q^{-1} - E(E^TQE)^{-1}E^T$ always positive definite?

If $Q \in \mathbb{R}^{m \times m}$ is symmetric and positive definite and $E \in \mathbb{R}^{m \times n}$ with $m > n$ has rank $n$, then is $Q^{-1} - E(E^TQE)^{-1}E^T$ always positive ...
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144 views

Finding the trace of matrix

If $K^T=K$, $K^3=K$, $K1=0$ and $K\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]=\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]$, how can I find the trace of $K$ and the determinant of $K$? I ...
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1answer
70 views

Gramian matrix test

Are there some test to know if a matrix $M$ is gramian ? $M$ is gramian if it exists a matrix W such $M=W^HW$. Also if it is possible to determine $W$. Thanks
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1answer
68 views

How to decompose an isometry $f\in O(\mathbb{R}^3)$ into reflections?

Consider a nondegenerate inner product space $(\mathbb{R}^3,\Phi)$. $O(\mathbb{R}^3)$ is the orthogonal group of $\mathbb{R}^3$. How do you decompose an isometry $f\in O(\mathbb{R}^3)$ into ...
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0answers
53 views

Behaviour of Hessian under coordinate change

According to this one source http://www.math.ethz.ch/~pinkri/Theses/2008-Bachelor-Andreas-Steiger.pdf, the Hessian of a function $F: \Bbb K^n \rightarrow \Bbb K$ should change under a coordinate ...
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230 views

Finding determinant for a matrix with one value on the diagonal and another everywhere else

Let us look the the matrix $\left(\begin{array}{ccccc} a & b & b & b & b\\ b & a & b & b & b\\ b & b & a & b & b\\ b & b & b & a & b\\ ...
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1answer
56 views

All Subspaces of $\mathbb{R^4}$ and $\mathbb{C^n}$ [Strang P129 3.1.14]

The subspaces of $\mathbb{R^n}$ are $\mathbb{R^4}$ itself, three-dimensional planes $\mathbf{n \cdot v = 0}$, two-dimensional subspaces $\mathbf{n_1 \cdot v = 0}$ and $\mathbf{n_2 \cdot v = ...
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580 views

Rank of the sum of two matrices

I have two square matrices $M$ and $N$ such that $M^2=M$ , $N^2=N$ and $MN=NM=0$. I'd like to prove that $\operatorname{rank} (M+N)=\operatorname{rank} (M)+\operatorname{rank} (N)$. I know that ...
4
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2answers
156 views

Diagonalization of $M=ab^t+ba^t$

Given $a=(a_i)_{i=1}^n$ and $b=(b_i)_{i=1}^n$ vectors in $\mathbb{R}^n$, we define the matrix $M=(m_{ij})_{i,j=1}^n$ as: $$ m_{ij}=a_ib_j + a_jb_i, $$ or equivalently $$ M=ab^t+ba^t. $$ Note that ...
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254 views

Free variables, nullspace for a matrix with the sum of certain columns = zero vector [Strang P142 3.2.20]

Suppose column 1 + column 3 + column 5 $\mathbf{ = 0} \quad (\bigstar)$ in a 4 by 5 matrix with four pivots. Which column is sure to have no pivot (and which variable is free)? What is the ...
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38 views

Isomorphism for invertible diagonal matrix

We know that the set of invertible diagonal $n\times n$ matrix over $\mathrm{GF}(q)$ ($\mathrm{DL}(n,q)$ is a subgroup of $\mathrm{GL}(n,q)$). Is there any isomorphism for $\mathrm{DL}(n,q)$? I think ...