Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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1answer
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Column/Row Space check

I have the following matrix: \begin{bmatrix} 1 & 2 & 0 & 1 & 0\\ 3 & 6& 1 & 6 & 1\\ 2 & 4 & -1 & -1 & -1\\ 4 & 8 & 0 & 4 ...
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0answers
11 views

How is this a substitution? Linear algebra transformation matrix misunderstanding

I found the following matrix equation in '3D Surveillance System Using Multiple Cameras', (authors: Ajay Kumar Mishra, Bingbing Ni, Stefan Winkler, Ashraf Kassima) (link here): I don't follow the ...
6
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2answers
259 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
4
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2answers
198 views

Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal

Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, ...
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0answers
36 views

Preparation for a Linear Algebra Class

I have just entered my Junior Year as a CS student. While I have already taken discrete math and Theory of Computation, and have not found myself needing any additional math skills thus far; I ...
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2answers
29 views

How does dot product work in matrix algebra?

I am working on a weighted minimization problem. Without the weights, the error function can be expressed as $e^T e$. With weights, $e$ first need to element-wise multiple by $w$, then the same ...
0
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1answer
39 views

How to remove vector x from (x'Ax)/(x'Bx)?

Are there any simple expression for the following scalar? $$a=(x'Ax)(x'Bx)^{-1}$$ where $x'=$transpose of $x$, $A,B\in\mathcal M_{n\times n}(\mathbb R)$ and $x\in\mathcal M_{n\times 1}(\mathbb R)$. ...
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3answers
73 views

Supose $A$ is a 4x4 matrix such that $det(A)=\frac{1}{64}$

Supose A is a 4x4 matrix such that $det(A)=\frac{1}{64}$ then $det(4A^{-1})^T$ I created a 2x2 matrix $B$ and transposed it both had the same deternminant I then found $det(B)$ and $det(B^{-1})$ ...
3
votes
7answers
192 views

Definition of characteristic polynomial

The question is very simple but it's giving me a hard time. I've been given the following definition for the characteristic polynomial of a linear transformation $$p_c(x)=\det(xI-T)$$ But what does it ...
2
votes
0answers
43 views

Density of Pythagorean triples

We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$. ...
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0answers
28 views

Find solutions to magic puzzle with sums

I need help to solve the folowing puzzle using linear algebra (matrix and Gauss-Jordan Method): (for example the second horinzontal line: w + w + w + z = 45 or the ...
2
votes
2answers
41 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
1
vote
1answer
26 views

How to avoid complex value for square root of a symmetric matrix?

I want to find square root of a matrix $Z$ which is a symmetric matrix using eigen values. So I find the eigenvalues($A$) and eigenvectors($B$) of $Z$ and find $B A^{1/2} B$. But because of small ...
9
votes
2answers
102 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
1
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0answers
37 views

Dual norm of the matrix L1 norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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0answers
40 views

Build up a not diagonalizable linear map

I need an hint for this problem. Let be $M = \begin{bmatrix}2 & 1 \\ -2 & 0\end{bmatrix} \in M_2(\mathbb{K})$ and $H=\{A \in M_2(\mathbb{K}) : AM=MA \} $ Build up a linear map $f: ...
2
votes
3answers
32 views

Vector spaces - Non-uniqueness of element with property of scalar-multiplicative identity element?

I am dabbling in vector spaces, thinking about the axioms on Wikipedia. Notably, $$1 \mathbf{v} = \mathbf{v},$$ i.e. identity element of scalar multiplication (IEOSM), attracted my attention. I am ...
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2answers
41 views

Calculate the unknown coordinates of a point $B (x_2,y_2)$ on a line with given distance from a known point $A(x_1,y_1)$

I have a line which represents a cross section. I have the coordinates of on its starting point. I need the coordinates of the end point of that cross section line. The distance between these two ...
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2answers
44 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
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1answer
27 views

Group Theory $Z_2$ representations

I am trying to understand some group theory. In the notes I am following, I am told: Recall the representations of $\mathcal{Z}_2$: Trivial: $\rho_0(e) = 1$, $\rho_0(a)$ = 1 (i) $\rho_1(e) = 1$, ...
4
votes
4answers
573 views

Finding eigenvalues and eigenvectors of an unknown matrix

This is given about a matrix A: $A \begin{bmatrix} 1\\2\\4 \end{bmatrix} =9 \begin{bmatrix} 1\\1\\1 \end{bmatrix} , A \begin{bmatrix} 1\\−3\\9 \end{bmatrix} =2 \begin{bmatrix} 1\\−3\\9 ...
0
votes
2answers
22 views

What does "$S\times S\rightarrow R$

In the textbook, Mathematical Methods and Algorighms for Signal Processing, Tood K. Moon, the $\mathbf{inner\;product}$ is defined it is a function $\langle\cdot,\cdot\rangle:S\times S\rightarrow R$ ...
0
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1answer
43 views

angle between two vectors-given in matrix form

Let $u=\left\{\begin{pmatrix} 1&a&0\\0&1&0\\0&0&1\end{pmatrix} \begin{pmatrix}1\\1\\0\end{pmatrix}:a\epsilon R\right\}$ $v=\left\{\begin{pmatrix} ...
4
votes
1answer
51 views

Adding a positive semidefinite matrix to a square matrix

Can one find some square matrix $A$ and a square positive semidefinite matrix $B$, such that the largest eigenvalue of $C=A+B$ is smaller than the largest eigenvalue of $A$?
0
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2answers
57 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
0
votes
1answer
320 views

Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
0
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0answers
46 views

prove that $\sum_{k=1}^\infty|x_k y_k|$ converges

Let $V$ be the space of real sequences $x_k$ so that $\sum_{k=1}^\infty x_k^2$ converges. Let $\langle x,y\rangle=\sum_{k=1}^\infty x_k y_k$ Prove that $\sum_{k=1}^\infty |x_k y_k|$ converges My ...
0
votes
1answer
60 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
1
vote
2answers
358 views

irreducible, diagonally dominant matrix

I am facing a problem for irreducible,diagonally dominant matrices. How to prove that irreducible, diagonally dominant matrix is invertible? Please help me in this problem.
1
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1answer
53 views

Why must $b=0$ for this linear system to have infinitely many solutions for all $a$?

Consider the parameterized linear system of equations represented by the augmented matrix: $$ \left[ \begin{array}{ccc|c} 1 & 0 & a & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & ...
0
votes
1answer
28 views

Construct and apply a rotation matrix by doing the following

Create a 2x2 rotation matrix $A \ne I$. Determine, showing all work, the location of point $(3, 2)$ when it is rotated using the linear transformation generated by the matrix. Also, demonstrate, ...
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2answers
22 views

If $B^T$ consists of a basis of $\mathrm{im} (A)^\perp$, then $\mathrm{im}(A)=\ker (B)$?

Well basically, the question is in the title: Suppose we have $A\in\mathbb{R}^{n\times m}$ with rank $d$ and we fix a basis $(b_1,\ldots,b_{n-d})$ of $\mathrm{im}(A)^\perp$. Let ...
0
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2answers
44 views

Subspace of matrices AB = BA

I'm stuck with the following exercise: Let A be an $n\times n$ diagonal matrix with characteristic polynomial: $$\prod_{i=1}^{k}(x-c_{i})^{d_{i}}$$ where $c_{1},...,c_{k}$ are distinct. Let $W$ ...
3
votes
1answer
40 views

How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?

So, I'm a newbie at complex vector math. I'm working on a 2D physics engine, and my issue is, with angular acceleration from torque, is it supposed to be positive or negative? I understand the right ...
0
votes
0answers
28 views

Updating the LU Factorization

I am looking for a way to update the $LU$ factorization of a general $m \times n$ matrix after adding a column to the matrix. I have to iterate this procedure so I will begin with a matrix that is $m ...
1
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2answers
27 views

Linear Transformation On Basis

What a Linear Transformation does on a basis? if the Linear Transformation is 1-1 and onto so every element of the basis goes to element of the basis of the other vector space? and what if it is not ...
6
votes
4answers
150 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
0
votes
3answers
56 views

Proof concerning matrix composition.

I have a statement which I don't know how to prove. All matrices are real, $n \times n$. For all $0 < k < n$ the following has to hold. It is impossible do define a matrix $A$ of rank ...
3
votes
2answers
92 views

Minimum linear subspaces cover problem

Given a set of vectors $V=\{v_1,v_2,...,v_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$ ($V_i$ may not be a subset of $V$), I want to find minimum number of sets from $\{V_1,V_2,...,V_m\}$, denoted as ...
0
votes
3answers
6k views

Non-trivial solutions implies row of zeros?

If there exist non trivial solutions, the row echelon matrix of homogenous augmented matrix A has a row of zeros. True or False? I'm not sure where to begin as to see why this would be true or ...
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0answers
20 views

A linear-algebraic property of stochastic matrices.

All matrices are real, $n \times n$. By a stochastic matrix, I mean any non-negative real matrix with rows summing to one. Denote the set of all stochastic matrices by $\mathcal{S}$. By $I_k$ I mean ...
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3answers
44 views

linear algebra-permutation

Given the permutation $$\sigma = \begin{pmatrix} 1&2&3&4&5\\3&1&2&5&4\end{pmatrix}$$ the matrix A is defined to be the one whose i-th column is the $\sigma(i)$-th ...
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1answer
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Canonical form 2nd order PDE

I want to reduce the following equation to canonical form $yu_{xx} + 2(x+y)u_{xy} + 4xu_{yy} = 0$ for $x > y > 0$ I chose ɛ to be $x^2 - \frac{y^2}{2}$ and η to be $2x - y$ Then I found the ...
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0answers
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Elementary proof, $A \to QAQ^T$ has same eigenvalues [duplicate]

Let $A\in \mathbb{R}^{n \times n}$ be a non singular matrix with $n$ eigenvalues and eigenvectors. How could one prove that $A$ has the same eigenvalues and eigenvectors as $QAQ^T$, where Q is an ...
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0answers
36 views

solving a system of 2 equations with 4 variables

Can someone show me how to solve this system of equations? \begin{align} 4X+12Y-7Z-20W&=22 \\ 3X+9Y-5Z-28W&=30 \end{align} Does this mean that there are two free variables here?
0
votes
2answers
31 views

Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
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0answers
17 views

How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$ $k(x, x') = (1 - 2|x-x'|)$ is a positive semidefinite function? It turns out to be psd function when ...
0
votes
1answer
39 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
12
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3answers
4k views

Importance of Linear Algebra

In one of his online lectures Benedict Gross comments that one can never have too much Linear Algebra. Also, looking around it seems like I can find comments to the effect that Linear Algebra has ...
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3answers
43 views

Find the Eigenvector of a matrix

Find the eigenvectors of the matrix $$\displaystyle\begin{bmatrix} 0 &2 &3 \\ -2 &0 &5 \\ -3 &-5 &0 \end{bmatrix}.$$ So I start with $|A-\lambda I|=0$ ...