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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Class of matrices for wich $A^T=J-A.$

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $$A = A^{\top}.$$ Instead, a matrix of ones or all-ones matrix is a matrix ...
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Right coordinates of a slanting line when slope is zero and left coordinates never changed after transformation

I have a line in a program I am developing that I want to remove the slant (slope to zero) then get the new coordinates after transformation that removes the slope. This is how the line with the ...
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22 views

Expanding vector norm into sum

I'm trying to expand a simple Euclidian vector norm into a sum of $x_i$ coefficients, so that for each $i$ term, I can treat everything as coefficients for a quadratic. I think I must have messed up ...
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1answer
40 views

The MU-puzzle from GEB

The MUI system only uses the three letters M,U,and I to make strings. The system has four rules that allow you to make new strings out of existing strings by manipulating them. Rules 1 and 2 lengthen ...
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4answers
515 views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
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2answers
28 views

Reduced row echelon form without introducing fractions at any intermediate stage

How can I reduce this matrix to reduced row echelon form but without using fractions in intermediary steps (I can use them in elementary row operations just not in the results in the matrix) $$ ...
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1answer
79 views

Group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
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45 views

Not sure if my Math Operation right or wrong. [on hold]

Hello guys I'm stuck with this. I'm not sure if my math operation right or wrong(Sorry for my bad English) Equation located here http://1.1m.yt/Cl3lGSbEn.png m=z=>>0?(z1-z2%50):(z1-z2%50)-50
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4answers
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proof that every finite matrix has an annihilating polynomial

I don't quite understand the proof my notes gave me. Dimension of $n$ by $n$ matrix is $n^2$. Hence if $k \geq n^2$ then $\mathbf{ \{ I, A, A^2, ..., A^k \} }$ is linearly dependent. So, there exist ...
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2answers
33 views

Proving existence and uniqueness of a matrix,

Let A be nxn with real coefficients and assume that it has n distinct eigenvalues, and all eigenvalues are positive real numbers. Let k $\ge$3 be an odd integer. a) Prove there exists a unique real ...
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52 views
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Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, $b\in\mathbb{R}^n$, $k>0$, and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive function. Consider the system of nonlinear ...
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1answer
54 views

Does it make sense to talk about complex matrices over the field of real numbers, R?

I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers. But I wanted to ask, ...
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1answer
29 views

Determinant proof question.

Using determinants, prove that if $A_1,A_2,...,A_m$ are invertible $nxn$ matrices, where $m$ is a positive integer, then $A_1A_2...A_m$ is an invertible matrix. Need help starting the proof. Do I ...
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2answers
39 views

Determinant Question.

Show that if $A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$, then $\det(A)=\frac{1}{2}\det\left(\begin{bmatrix}1 & 1\\tr(A^2) & (tr(A))^2\end{bmatrix}\right)$. I tried finding the ...
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1answer
31 views

Determining kernel and image of linear map

Problem: Which of the following maps are linear? Determine the kernel and the image of the linear maps and check the dimension theorem. Which maps are isomorphisms? 1) $L_1: \mathbb{R} \rightarrow ...
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2answers
27 views

Matrix multiplication and determinant question

Show that if $\det(\begin{bmatrix}b & c\\a & b\end{bmatrix})=0$ with $A=\begin{bmatrix}a & a\\b & b\end{bmatrix}$ and $B=\begin{bmatrix}b & b\\c & c\end{bmatrix}$ then ...
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2answers
42 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
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2answers
32 views

Block Matrix Zero Determinant Implication?

Recently I've been working with a number of square (order of 2n) matrices whose determinants are zero. That is, $$\det\begin{bmatrix}A&B\\C &D \end{bmatrix} = 0$$ where each of A,B,C, and D ...
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3answers
153 views

Linear dependence of these functions?

How can I check if these three functions (which belong to vector space $R^R$) are linearly dependent: $$e^{2x}, e^{3x}, x$$ If I take $\alpha, \beta, \gamma ∈ R$ and write the linear combination as: ...
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0answers
9 views

How to build an 2-D polynomial from 1-D orthogonal polynomials

I have an set of orthogonal polynomials such as I want to build an 2D polynomial following the equation $$P_k(x,y)=P_k(x)P_k(y)$$ where $k=1..4, (x,y) \in [-1, 1]^2$ Based on given $P_n(x)$ as ...
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1answer
26 views

Which polynomial has similar properties with Legendre?

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? ...
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1answer
12 views

dimension of the quotient of a bialgebra

I am stuck in a proof of a lemma that I am in need of. The situation is as follows: Let $k$ be a field and $A$ and $B$ two finite-dimensional $k$-bialgebras, where the dimension of $A$ is a prime ...
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3answers
49 views

Let $v_{1}=(1,-2,3),v_{2}=(0,-1,2)$. Enlarge $\{v_{1},v_{2}\}$ to a basis for $\mathbb{R}^3$.

For instance, let $v_{1}=(1,-2,3),v_{2}=(0,-1,2)$. The set $\{v_{1},v_{2}\}$ is linearly independent. Enlarging $\{v_{1},v_{2}\}$ to a basis for $\mathbb{R}^3$ I simply form a matrix using ...
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1answer
13 views

How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
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1answer
22 views

Variance of subset vs total variance

Is it true that the variance of subset is smaller than variance of the total set? Given each element in the set is a N-dimensional vector, and the distance is defined as Euclidean distance. Variance ...
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0answers
21 views

Rank of the sum of two rank 1 matrices, proof check

Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that $uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$. That is, for any vector $u$ of ...
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0answers
15 views

Put, in matrix form: $t_i=\sum_{j=1}^n \frac{w_j-w_i}{1+e^{-(x_i-x_j)}}$, $\forall i=1,2,…,n$

I have the set of equalities $$t_i=\sum_{j=1}^n \frac{w_j-w_i}{1+e^{-(x_i-x_j)}}, \ \ \forall i=1,2,...,n$$ and I try to write them in a more concise form. I tried to do so: $$t_i=\sum_{j=1}^n ...
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1answer
32 views

Solutions to the equation $a + b - ab/t = t/2$

$$a + b - ab/t = t/2$$ Where $0 < a < b < t$, $a,b,t \in \mathbb{N}$ and t is even, ie $t\mod2 = 0$ What are the possible values for a, b for a given t? For example, if t = 1000, then a = ...
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is this conclusion true or false?

Let $\mathcal{A}$ be a factor Von Neumann algebra and $\Phi$ is a map on $\mathcal{A}$ which is injective and surjective and $\Phi(0)=0$. If $A, B, C \in \mathcal{A}$ and ...
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0answers
10 views

Radial Basis Function on 2dimensional data

my data is a 2 dimensional point:x=(x1,x2). i want to apply RBF kernel on my data and transform it to infinite dimension space. i know what will be its infinite dimension form when data is defined in ...
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1answer
24 views

Why is $[\alpha]_{\mathfrak{B}}=P[\alpha]_{\mathfrak{B'}}\rightarrow\alpha'_{j}=\sum_{i=1}^{n}P_{ij}\alpha_{i}$ obvious?

In the middle of looking into one of the theorems regarding coordinates a part of the proof of the one that I was reviewing at that time—which is presented below—puzzled me in that it was not so ...
3
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1answer
29 views

Find the inverse of a specific Vandermonde matrix

Let $$ V=\begin{bmatrix} 1& 1& 1& \cdots& 1 \\ 1& \xi& \xi^{2}& \cdots& \xi^{n-1} \\ 1& ...
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1answer
21 views

The Maximum Eigenvalue of $F\mathrm{max(B)}F^T - FBF^T$

$F$ is a $b \times n$ real matrix. $B$ is a $n \times n$ real matrix, constructed by $B = w^T w$, where $w$ is a row vector with strictly positive real numbers, and clearly $B$ is a rank 1 matrix. ...
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1answer
31 views
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how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
1
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1answer
28 views

Proving that $V = U_1 \oplus U_2 \oplus \ldots \oplus U_k$.

Problem: Let $V$ be a vectorspace and $\beta$ a basis for $V$. Now make a partition of $\beta$ in a disjoint union of subsets $\beta_1, \ldots, \beta_k$ and let $U_i = \text{span}(\beta_i)$ for every ...
3
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0answers
56 views

Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$.

I want to prove the following statement: Let $A \in \Bbb R^{n\times n}$ with eigenvalues $\lambda$ and eigenvectors $v$. Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$. ...
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1answer
35 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
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1answer
43 views

Restriction of a linear algebra to an affine subspace?

Let's assume $V$ and $W$ are finite dimensional vector spaces and, $F:W\longrightarrow V$ is a one-to-one affine map i.e, $F(W)$ is an affine subspace of $V$. Also, let $T:V\longrightarrow V$ is a ...
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Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
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Find 3 eigenvectors and rife vectors of rank 1 matrix

I am trying to solve this without have to factor a polynomial of degree three of higher. $$ \begin{pmatrix} 2 & 4 & 2 \\ 4 & 2 & 4 \\ 2 & 4 & 2 ...
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Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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1answer
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solve problems using linear systems [closed]

Leanne works at a greenhouse store. she needs to plant a total of 32 bulbs. two types of bulbs are available. she is asked to plant 3 times as many crocus bulbs as tulip bulbs. how many of each should ...
2
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2answers
36 views

Kernel of a bounded linear operator on a normed linear space need not be closed or open?

How should be the kernel of a bounded linear operator on a normed linear space as a set? Kernel of a bounded linear operator on a normed linear space need to be closed or open? Or it need not be ...
2
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1answer
41 views

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row?

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row(uniform distribution)? What sort of algorithm should I use to do this task? Brute Force algorithm- ...
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1answer
48 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
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4answers
263 views

If a matrix has positive, real eigenvalues, is it always symmetric?

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric? A class of symmetric ...
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2answers
30 views

Express $x+y+z$ in terms of $a$ and $b$ [closed]

If $A = X + Y$ and $B = X + Z$, find the value of $X+Y+Z$ in terms of $A$ and $B$.
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1answer
62 views

Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
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1answer
113 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
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5answers
30 views

Find a matrix which maximizes expression

Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$ A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy $$ How can I find the matrix $A$ that maximizes expression ...