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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If A is 5 by 3 and B is 3 by 5 (with dependent columns), Is $AB = I$ impossible?

Let me first introduce the problem. This is part of the quiz problem from MIT's 18.06 course (Spring 2012 semester, quiz 1, problem 3). My question is related to (b) but (a) is mentioned in the ...
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1answer
22 views

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$.. [on hold]

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$ dimensional subspace of $\mathbb{R}^n$ that's contained in $H$, show that $G = H$ I know that a subspace of $\mathbb{R}^n$ is ...
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0answers
10 views

Solutions to Binary Equations

Let $A\in M(m,n,\{0,1\}$) (i.e. $m \times n$ matrices with entries in $\{0,1\}$) and $x,y\in \{0,1\}^n$. We will denote the $i$-th row of $A$ as $row_i(A)\in \{0,1\}^n$. Define, $ z_i = \begin{...
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1answer
564 views
3
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1answer
62 views

Is every totally ordered finite dimensional vector space a lexicographic order for some basis?

Let's say we have a finite-dimensional vector space $V$ over a totally ordered field $\mathbb{K}$. Is every choice of totally ordered vector space structure (i.e compatible with the addition and ...
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1answer
2k views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
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4answers
41 views

What is the calculation behind this linear algebra basic question?

$$ \begin{bmatrix}1&2\\3&8\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}\ \text{yields}\ \begin{bmatrix}1&2\\0&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\...
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2answers
74 views

How come two of the eigenvalues are same?

Question is about finding the eigenvalues of the matrix : $$\begin{bmatrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{bmatrix}$$ the matrix would become $$\begin{bmatrix} -...
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1answer
18 views

Interchanging vectors coordinates

Is there any relation between two vectors with interchanging coordinates .. i.e: the x component of the first is the y component of the second and vice versa.
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28 views

Answer checking - involved derivative under summation

I'd like someone smarter and more experienced than me to check my answer and give advice on how to do it better and derive a closed form for what I'm looking for. Given a matrix $Y \in \mathbb R^{m \...
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2answers
41 views

The first 'primordial' basis of a finite vector space

Let's take a vector space $V $ and set $V= \mathbb{R}^3 $ for ease of mind. Usually we equip $V $ with the standard basis $\{e_1,e_2,e_3\} $ and we express all our vectors in that basis: $$v = (a,b,...
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Reducing row of a matrix with 1 parameter

I have this matrix: $$\begin{bmatrix}a-2 & 2 & 3 & 0 & -1\\ 0 & 1 & 1 & -1 & 0\\ 0 & 0 & 2(1+a) & 1+a & 3\end{bmatrix}$$ I need to find what values ...
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2answers
23 views

Number of multisets with restrictions on specific element count

I am looking to find the number of multisets with restrictions on the number of specific elements. This isn't for homework, it is a work related problem. My set of items is {A, a, B, b}. I want to ...
2
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1answer
22 views

$K$-algebra homomorphism

Let $\mathbb{T}_n(K)$ the set of all triangular matrices, where $K$ is a field. And let $U$ the set of all matrices $\lambda=[\lambda_{ij}]$ in $\mathbb{T}_n(K)$ with $\lambda_{ii}=0$. I want to ...
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0answers
13 views

Is there a correct way to compute matrix K, for SVD, and eigenvalue analysis

I'm reading two different papers on multivariate data analysis, both use the matrix $K$ for singular value decomposition, eigenvalues and eigenvectors. This is the case where there are two matrices, ...
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3answers
10k views

Prove that every positive semidefinite matrix has nonnegative eigenvalues

There is a theorem that every positive semidefinite matrix only has eigenvalues $\ge0$. How can i prove this theorem?
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1answer
25 views

About the eigenvalues of a block Toeplitz (tridiagonal) matrix

I have found the following $n\times n$ squared matrix in one stability analysis problem (i.e. I have to identify the sign of its eigenvalues) $$ A(\theta) = \begin{bmatrix} W(\theta)+W(\theta)^T & ...
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3answers
113 views

Calculation of $\frac{a_{20}}{a_{20}+b_{20}}$?

The solution of $\frac{a_{20}}{a_{20}+b_{20}}$ is $-39$ (This is wrote by answer sheet) from the recursive system of equations : \begin{cases} a_{n+1}=-2a_n-4b_n \\ b_{n+1}=4a_n+6b_n\\ a_0=1,b_0=0 \...
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2answers
22 views

(Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
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2answers
871 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 ...
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1answer
92 views
+100

Determinant of matrix with non-invertible blocks

I am trying to find a nice way of computing the determinant of the matrix \begin{equation} M= \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb{R}^{T\times T} \end{equation} where $A \...
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2answers
26 views

The Matrix of a reflection (around abitrary plane)

Let $\Upsilon :\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a reflection across the plane: $\pi : -x + y + 2z = 0 $. Find the matrix of this linear transformation using the standard basis vectors and the ...
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0answers
25 views

Solving Systems of linear equations between a square matrix and a rectangular matrix with block decomposition

I am trying to decompose solving a system of linear equations using block decomposition where I have an (n x n) matrix A, which is a lower/upper triangular matrix, and a matrix B, which is a ...
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0answers
30 views

Generate function from data

I have a series of inputs and outputs : Inputs -> Outputs 1,2,3 -> 4 4,5,6 -> 5 7,8,9 -> 6 Is there a field of study that can generate a single ...
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0answers
12 views

How to linearize two discrete maps with time delay feedback

I have a 2-D system of two discrete maps $x_{n+1} = f(x_n) + P_1(y_n - y_{n-1})$, $y_{n+1} = g(y_n) + P_2(x_n - x_{n-1})$ with $g,f$ being smooth functions and $P_1, P_2$ belonging to the reals ...
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0answers
28 views

prove that $A^{-1} = (1/detA) \operatorname{cof} A^T$ [on hold]

Can you please explain to me how to prove this theorem? Theorem: if $\det(A)\ne 0$, then $A$ is invertible and $A^{-1} = \frac 1{\det(A)} \operatorname{adj} A$
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1answer
24 views

Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...
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1answer
41 views

How to project $x_2$ onto $u_1$

I'm following a solution from here (the first problem), I don't understand how to "project $x_2$ onto $u_1$" 1) how does:$\begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix}$ ...
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0answers
11 views

when the k-algebra product is a vector [on hold]

I have a question, maybe stupid because I'm physics student and so I'm not following algebra, tensor calculus courses. I hope to clarify well. as it was written on page 323 in this book a product on ...
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3answers
49 views

If $T$ and $T^2$ have equal rank then $V=\ker T\oplus {\rm im}\, T$ for $V$ finite dimensional.

I am trying to prove the following: Let $V$ be a finite-dimensional vector space. Consider an operator $T$ on $V$ such that $\text{dim range}(T)=\text{dim range}(T^2)$. Show that $V=\text{null}(T)\...
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1answer
39 views

Show that the trace of A is less than n

Let $A$ be an $n\times n$ matrix with complex entries such that $A^k=I_n$ for some positive integer $k$. Show that the trace of $A$ satisfies $$|tr(A)| \leq n.$$ I have no idea how to approach this ...
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1answer
17 views

How to prove or disprove there is a unique solution to this linear system where the variable are constrained to the interval [0,1].

Be $m$ and and $n$ integers such that $1\leq m \leq n$. Be $l_k$ variable with $dom(l_k)=[0,1]$. Take the folowwing linear equation system: $\sum_\limits{k=1}^{n} l_k =1$ $\sum_\limits{k=1}^{n} k\...
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1answer
28 views

Overlap between two vectors

Given are two vectors ${\bf g}_1, {\bf g}_2\in\mathbb{R}^N$ with non-zero scalar-product ${\bf g}_1^\top{\bf g}_2 \ne 0$. Then there exist three unique orthogonal unit vectors ${\bf e}_1, {\bf e}_2, {\...
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1answer
23 views

Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
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0answers
36 views

Finding a linear system to solve quadratic equations

considering an equality with a polynomial of second degree where the coefficient for $x^2$ is $1$ I know that $$ a x^2 + b x + c = a(x-\alpha)(x-\beta) = 0 $$ I also know that $$ \alpha + \beta = -...
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1answer
18 views

Construction of a Module isomorphism

Let $R$ be a PID. Consider the sets $X_0=\{v_0,v_1,v_2\}$ and $X_1=\{e_1,e_2,e_3\}$ and let $C_i$ be the free $R$-module on $X_i$ for $i=0,1$. Consider the $R$-module homomorphism $$C_1\;\...
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0answers
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Conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z}) × GL_m(\mathbb{Z}/q\mathbb{Z})$ [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z})× GL_m(\mathbb{Z}/q\mathbb{Z})$ of cyclic subgroups of order pq?.
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0answers
29 views

How to randomly generate two integer matrices $A$ and $B$, so that entries of 3 metrics $A$, $B$, and $AB$ are within certain range?

I ran into this question when writing a program. I need to generate two matrices, and calculate their product. However, I must ensure all entries are within 8-bit signed integer range, i.e. $[-128, ...
2
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1answer
547 views

MATLAB determining elementary matrices for LU decomposition

I am confused by this question I am studying for MATLAB practice.
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0answers
16 views

Water drop evaporation time and contact angle

I'm measuring water drop evaporation on different surfaces and it would be nice to have an equation to roughly estimate evaporation time (or contact angle). Some drops are hydrophobic, others ...
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0answers
35 views

What is a linear isomorphism?

I am working with the book Manifolds and Differential Geometry from Lee and I am a little bit puzzled since he sometimes talks about linear isomorphism (proposition 2.3 for example). But isn't an ...
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1answer
55 views

Prove that the ellipsoid $x^T W x \leq 1$ is invariant under $f (x) = A x$ [closed]

Given matrix $W \succ 0 $ and a set $\mathcal{Z} := \{z \mid z^T W z \leq 1\}$, prove that if $Az \in \mathcal{Z}$ and $z \in\mathcal{Z}$, then the following inequality holds $$ A^T W A - W \...
3
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1answer
379 views

Linear system with positive semidefinite matrix

I have a linear system $Ax=b$, where $A$ is symmetric, positive semidefinite, and positive. $A$ is a variance-covariance matrix. vector $b$ has elements $b_1>0$ and the rest $b_i<0$, for all $...
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1answer
75 views

Is it true that $SL(n, \mathbb R)=<\{ABA^{-1}B^{-1} : A,B \in GL(n,\mathbb R) \}$ >? [closed]

Is it true that $\operatorname{SL}(n, \mathbb R)=\left\langle \left\{ABA^{-1}B^{-1} : A,B \in \operatorname{GL}(n,\mathbb R) \right\} \right\rangle$?
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1answer
26 views

Basic questions about optimizing concave function with constraints

Consider the following problem: \begin{align} {\tt Maximize} \quad M(\mathbf y)& = \log \Big(\prod_i U_i(y_i) \Big) \\ y_i & = \sum_{j=1}^{m} \frac{x_{ij}}{a_{ij}} \\ \sum_{i=1}^{n} \frac{c_i ...
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0answers
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Curious about the deduction procedure on shannon transform and stieltjes transform

I found some scholars clarified that the relationship between the Shannon transform and Stieltjes transform with \begin{equation} \frac{\gamma}{\log e}\frac{d}{d\gamma}\mathcal{V}_N(\gamma) = 1 - \...
2
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1answer
31 views

Least Squares Algorithm with Inverse Norm

Given an overdetermined linear system $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m \times 1}$ with $A < 0$ and $b < 0$. What is a good way to numerically determine $$ \min_x \left\lVert ...
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0answers
39 views

what is Expected Mean

Thus the expected mean $\mu$ of the set $\mathcal S$ can be given as \begin{align*} \mathbb E \mu&= \sigma^2+\frac 1r \sum_{i=1}^m\left(\mathbb E\lambda_i-\sigma^2\right)\\ &\geq \sigma^2+\...
2
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1answer
32 views

Proof Determinant of Block Matrix does not depend of a variable

I have the following matrix (called BRM): $ BRM = \begin{bmatrix} -A & A & \mathbb{0}_{3\times3} & B & \mathbb{0}_{3\times1} & \mathbb{0}_{3\times1} \\ -C &...
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0answers
36 views

Real and imaginary part of tensors of matrices

Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...