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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How come associative law of matrix multiplication won't work when permutation matrices come in. Which is the case for some

if $$x=y$$ explain why $$Px=Py$$ I believe this part is very since when we do $$P^{-1}Px = P^{-1}Py$$ from here $$x=y$$ But the other part of the question seems much more confusing then $$(Px)^...
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1answer
8 views

Are a uniformly random polynomial's roots are distributed uniformly in the field?

Assume we have a $\mathbb{F}_p$, where $p$ is a large prime (e.g. 128-bit value). We define all polynomials over the field, and pick a polynomial,$P(x)$, of degree $d$, where the polynomials' ...
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3answers
44 views

How to prove $I-BA$ is invertible

Show that $I-BA$ is invertible if $I-AB$ is invertible. And also, we have to prove that eigen values are same for $AB$ and $BA$ Till now, i used the equation $(I-AB)(I-AB)^{-1}=I$ which gives $(I-AB)...
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0answers
7 views

Eigenvalue of multiplicity k of a real symmetric matrix has exactly k linearly independent eigenvector

If A is an nxn real symmetric matrix then A is diagonalisable. In other words, If A is a symmetric nxn matrix, then there exists an orthogonal matrix $P$ such that $P_{-1}AP=P_{T}AP=D$, a diagonal ...
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8 views

alignment of two sets of vectors

I have a maximisation problem to do with aligning two ordered sets of 3D unit vectors. I want to apply the same rotation to all the vectors in one set so that they are in closest alignment with those ...
2
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1answer
56 views

Problem about linear algebra [duplicate]

Suppose we have two $n \times n$ square matrices A and B such that $AB=BA$. It is known that A, B and AB all have n distinct eigenvectors that is a basis of $\mathbb{C}^n$. Can we then show that there ...
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Discussing the solutions of a system of equations

I have a question regarding the following system of equation: $$ \begin{cases} ax+by+cz+dw=\eta_1\\ ax^2+by^2+cz^2+dw^2=\eta_2\\ ax^3+by^3+cz^3+dw^3=\eta_3\\ ax^4+by^4+cz^4+dw^4=\eta_4\\ ax^5+by^5+cz^...
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3answers
23 views

How to find other basis of polynomials of degree three or less?

How can i find a basis of polynomials of degree three or less, which is other than $\{1,t,t^2,t^3\}$ ?
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2answers
17 views

Finding inverse by elimination

Find the inverse of the matrix $A$ below by elimination on [A I] By expanding the matrix into an alternating matrix. $$ A= \begin{bmatrix} 1 & -1 & 1 & -1 \\ 0 & 1 & -1 & 1 \\ ...
2
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1answer
542 views

MATLAB determining elementary matrices for LU decomposition

I am confused by this question I am studying for MATLAB practice.
2
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1answer
53 views

Solving the linear system $XL + L^TX = M$ efficiently

I'm wondering of an efficient way to solve the following system for the symmetric matrix $X$, given a positive semi-definite matrix $S$ and any matrix $M$: $$ LL^T = S $$ $$ XL + L^TX = M $$ $$ (XL) + ...
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13 views

is I both a lower triang enad upper triangle ( Also proving L1=L2 )

First part of the question is $$ A= L_1D_1U_1\\ A = L_2D_2U_2\\ Prove\\ L_1= L_2\\ D_1 = D_2 \\ U_1 = U_2 \\ $$ My attempt seems correct but not quire sure whether it's mathematically constructed. $$...
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2answers
18 views

Choosing independent entries in a symmetric matrix

So, the question is how many entries can be chosen indepently in a symmetric matrix of order n? 2) How many entries can be chosen indepently in a skew-symmetric matrix $$ K^T=-K $$ of order n. The ...
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3answers
46 views

Showing A is not invertible

$$ A= \begin{bmatrix} 2 & 1 & 4 & 6 \\ 0 & 3 & 8 & 5 \\ 0 & 0 & 0 & 7 \\ 0 & 0 & 0 & 9 \\ \end{bmatrix} $$ We are asked to show A is not invertible ...
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0answers
22 views

Workability of linear equation solving methods for different fields?

So far, I have mostly done linear algebra over $\mathbb R$ and $\mathbb{C}$. I know there exist very many methods to solve equation systems for those fields, of which a few are Gaussian Elimination, ...
3
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1answer
28 views

Struggling with the “Umbral Algebra”

I am reading a lot of papers and material that deal with "Umbral Calculus" and "Umbral Algebra" and am finding myself very confused and am hoping I can get a little more clarity. For those willing to ...
3
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1answer
34 views

Is there an $\alpha\in\mathbb{R}^m$, such that $\alpha_i > 0$ and $A\alpha\in S$?

$A$ is a real $n\times m$ matrix and set $S\subseteq \mathbb{R}^n$ is defined as $$S = \{(x_1,\dots, x_n)\in \mathbb{R}^n\mid \forall(i,j)\in I.\; x_i< x_j\}\text{,}$$ where $I$ is a possibly empty ...
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0answers
27 views

Vector space complement to a multiplicatively closed subspace is an ideal

Let $V$ be a vector space over $\mathbb{C}$ of any dimension and suppose we have an associative multiplication $V \times V \to V$ making $V$ into a commutative ring with unity. Let $V=U \oplus W$ be a ...
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2answers
15k views

Understanding rotation matrices

How does $ {\sqrt 2 \over 2} = \cos (45^\circ)$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? ...
3
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1answer
53 views

Maximization of a determinant

I'd like to compute $$ \DeclareMathOperator*{\argmax}{arg\,max} A^*=\argmax_{\substack{A\in\mathbb{R}^{d\times k}\\A^T A=I}} \det(A^T \Lambda A) $$ where $k\leq d$, $\Lambda=\operatorname{diag}(\...
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1answer
1k views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 = ...
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2answers
276 views

Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements

I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is not diagonal or triangular. Is there a term for such matrices, and have they been researched?
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31 views

Notationquestion: Matrices

I have to determine the base change matrix $S=M(id,A,B)$. Now i looked at how to do it, but which base is right and which left in doing it? i would write A|B and then try to get the elementary ...
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1answer
28 views

If both of $A,A^{-1}$ have entries from non negative integers then can we say $A$ is a permutation matrix?

I've shown if both of $A,A^{-1}$ (assuming $A$ to be invertible) are $n\times n$ matrices with entries from natural numbers then both of them have to be permutation matrices. Now my question is if ...
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2answers
36 views

Finding a point between 2 moving points colinearly , given 2 moving points and distance. [on hold]

A------B---------------C A and C are moving points that can move anywhere A = (xa, ya), C = (xc, yc) B (xb, yb) is a point between A and C colinearly With one condition that distance of AB = ...
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2answers
774 views

Why is some power of a permutation matrix always the identity?

If you take powers of a permutation, why is some $$ P^k = I $$ Find a 5 by 5 permutation $$ P $$ so that the smallest power to equal I is $$ P^6 = I $$ (This is a challenge question, Combine a 2 ...
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1answer
52 views
+50

Rank and null space of a particular block matrix.

Let $D_1, D_2 \in \mathbb{R}^{N \times N}$ be diagonal matrices with diagonals that are linearly independent vectors. Let $A, B \in \mathbb{R}^{N \times N}$ be rank-deficient matries. Define $S = \...
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1answer
10 views

Proof: $x_{1-n}$ linear dependant, w alternating multilinearform $\Rightarrow$ $w(x_1,…,x_n)=0$

Let $F$ be a field and $X$ a $F$-linear Space with $dim_FX=n\in\mathbb{N}$. Let $w$ be an alternating multilinearform on $X$ and let $x_1,\cdots ,x_n\in X$ be linear dependant. Show that $w(x_1,\...
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0answers
7 views

Eliminate asymptote using projective transform

I have a well-behaved curve $f:\mathbb{R}\rightarrow \mathbb{R}^2$ which has exactly one linear asymptote passing through points $p$ and $q$ in $\mathbb{R}^2$. I would like to find a projective ...
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1answer
16 views

Eigenvalues of product of p.d. Matrix with upper-triangular Matrix

Let $A$ be a positive definite matrix (positive eigenvalues). Let $B$ be an upper triangular matrix, with ones in its main diagonal (i.e. all its eigenvalues are 1). Is there anything I can say about ...
4
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1answer
41 views

Prove a Hermite polynomial property by linear algebra

Let $$ X=\begin{pmatrix} 0 & & & &\\ 1 & 0 & & &\\ & 1 & 0 & &\\ & & \ddots & \ddots &\\ & & & 1 & 0 \end{pmatrix}, D=\...
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1answer
438 views

Edge weight function for graph instance of scheduling and allocation problem

I have difficulties developing a proper (non-scalar) edge cost function $c_e$ for my resource scheduling problem, which I mapped into a graph problem. Processes $P_i$ need resources $R_i \in \mathcal{...
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0answers
29 views

What is the Linear space $R^{2 \times 2}$ means visually in linear algebra?

Why mathematicians decided to define a space of matrices if it does not make any sense visually? What are the uses of such linear space? and why my professor tells me that matrices can be written ...
0
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1answer
21 views

Find $P$ such that $A=P^{-1}JP$ where $A$ is the matrix of $f$ and J is the Jordan Form. $P$ non invertible?

Find the Jordan Form and a basis of Jordan for the endomorphism of $R^4$ $$f(x,y,z,t)=(x,x+y-t,-2x+y+z+2t,-x+2t)$$ After doing all the process, I find $P$ such that $A=P^{-1}JP$ where $A$ is ...
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2answers
35 views

If $A$ is diagonizable then $p(A)$ is diagonalizable

Show that if a matrix $A$ of size $n \times n$ is diagonalizable, then $p(A)$ is diagonalizable for each polynomial $p$.
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0answers
26 views

First steps in derivation of matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
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2answers
64 views

what is the difference between cross product and exterior product?

I have learn that the exterior product is an oriented plane called bivector given as $A \times B = |A||B| \sin x (i \times j)$ For $x \in(-\pi,\pi)$. I will like someone to derive the cross product ...
0
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1answer
31 views

Transformations between coordinate frames

Suppose I have three coordinate frames: $A$, $B$ and $C$, all in 2D space. In homogeneous coordinates, I deduce, by inspection, the transformation matrices between each of these ($T_{AB}$, $T_{BC}$ ...
3
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1answer
2k views

Java Tetris - Using rotation matrix math to rotate piece

I'm working on building tetris now in Java and am at the point of rotations... I originally hardcoded all of the rotations, but found that linear algebra (matrix rotations) was the better way to go. ...
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0answers
28 views

Show that $A\varphi_j=\left<A\varphi_j,\varphi_j\right>\varphi_j$ and $A^*A\varphi_j=s_j(A)^2\varphi_j$ for all $j$

Let $A$ be a bounded linear (compact) operator acting op a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\...
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$s_j(A)=1$ for all $j$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
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1answer
39 views

Existence of a basis $B$ such that $M(\phi,B)=E$

Considering the matrix $$A=\begin{bmatrix}{2}&{1}&{0}\\{1}&{0}&{-1}\\{0}&{-1}&{-2}\end{bmatrix}\in M_3(R).$$ And $\phi:R^3 \times R^3\longrightarrow R$ the bilinear ...
2
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2answers
750 views

calculate centroid of triangle on a graph

Given ANY three points on a graph that form a triangle, how do you find the centroid using geometry? So basically I have three points (X1, Y1), (X2, Y2), and (X3, Y3). I am trying to use the slopes ...
2
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0answers
19 views

Action of the Product of Two Linear Functionals on a Polynomial

I am looking for help with the following problem. Here we denote the action of a linear functional on a polynomial by $$\langle L\mid p(x)\rangle$$ Suppose that there are two linear functionals $...
0
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1answer
12 views

The map $f$ is degenerate or non-degenerate?

Let denote by $M_{3,2}(\mathbb C) $ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
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1answer
13 views

Orthonormal Basis /Linear Combination

(a) Find an orthonormal basis {${v_1,v_2,v_3}$} of the image of the linear function given by the matrix $$A=\begin{pmatrix} 1 & 1 & 2 \\-1 & 0 & 0 \\ -1 & 0 & 1\\ 0 & 1 &...
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2answers
21 views

How to prove Binary field is a vector space? [on hold]

How can I prove that the binary field is a vector space over that field? Thank you
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1answer
26 views

Solving vector equation 2

Using vector method, show that the vector equation $$\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$$ is satisfied if $$\bar{x}=\lambda \bar{a}+\bar{a}\times \frac{\bar{a}\times (\bar{d}\...
3
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1answer
27 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...