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 ...

learn more… | top users | synonyms (1)

0
votes
2answers
23 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 ...
0
votes
0answers
12 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 ...
0
votes
0answers
24 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 ...
1
vote
0answers
6 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 ...
0
votes
0answers
23 views

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

Can you please explain to me how to prove this theorem? Theorem: if $\det(A)\ne 0$, then $A$ is invertible and $A^{-1} = \frac 1{\det(A)} \operatorname{adj} A$
0
votes
1answer
23 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, ...
1
vote
1answer
36 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:$\begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix}$ ...
0
votes
1answer
8 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 & ...
0
votes
0answers
10 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 ...
3
votes
3answers
45 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)\...
1
vote
1answer
37 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 ...
0
votes
1answer
16 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\...
1
vote
1answer
27 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, {\...
0
votes
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 $w_2$ ...
1
vote
0answers
32 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 = -...
0
votes
1answer
16 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\;\...
-4
votes
0answers
22 views

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?.
1
vote
0answers
27 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
votes
1answer
547 views

MATLAB determining elementary matrices for LU decomposition

I am confused by this question I am studying for MATLAB practice.
0
votes
1answer
58 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} -...
0
votes
0answers
15 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 ...
1
vote
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 ...
1
vote
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
votes
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 $...
1
vote
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 $\operatorname{SL}(n, \mathbb R)=\left\langle \left\{ABA^{-1}B^{-1} : A,B \in \operatorname{GL}(n,\mathbb R) \right\} \right\rangle$?
3
votes
3answers
100 views

Recursive System of Equations and one Solved Example in 2007 GATE Exam?

The solution of $\frac{a_{20}}{a_{20}+a_{20}}$ 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=1 \...
1
vote
1answer
24 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 ...
-1
votes
0answers
11 views

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
votes
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 ...
-2
votes
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
votes
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 &...
1
vote
0answers
35 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 ...
0
votes
1answer
19 views

Calculating the missing two points of rectangle if 2 points and the aspect ratio are known

How can I calculate the missing two points of a rectangle if I know 2 points (top left and top right) and the aspect ratio i.e 16:10. For example: Top left: A(834, 449) and Top right: B(1675, 423)
1
vote
5answers
143 views

Good true-false linear algebra questions?

Can you suggest me a collection of true-false linear algebra questions, like the ones found in the MIT exams, if possible with solutions (i.e. explanations)? Sorry if it turns out that my request is ...
2
votes
1answer
15 views

Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
3
votes
3answers
69 views

Which non-negative matrices have negative eigenvalues?

It's easy to proof by counterexample that non-negative matrices can have negative eigenvalues. For example, the following matrix have -1 as an eigenvalue: $$ A = \begin{bmatrix} 0 & 0 & 0 ...
10
votes
4answers
305 views
+50

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
0
votes
1answer
17 views

Choosing the seed for a LFSR

I was just wondering how the seed of a LFSR is chosen and is there any connection between the seed chosen and cryptographic strength of the keystream generated? Thank you
0
votes
1answer
29 views

Question on proof of number of solutions of linear system

The proof my book uses starts off by saying: "If the system has exactly one solution or no solutions, then there is nothing to prove", and then continues on by assuming there is an infinite ...
1
vote
1answer
37 views

prove subspace of a vector space

Consider the subset $T$ of $\mathbb{R}^2$ defined as follows: $T := \left\{(x, y) : x, y \in \mathbb{R} : y = 3x \right\}$. Prove that T is a subspace of the vector space $\mathbb{R}^2$. My attempt: ...
1
vote
1answer
50 views

Solving linear inequalities with some variables subject to non-negativity constraints

I've a system of equations $Ax=0$ which is subject to the constraint that the first few components of the unknown vector x is non-negative (No constraint on remaining components of x vector). The $A$ ...
1
vote
2answers
35 views

Finding an orthonormal basis for the plane $x_1 - 5x_2 - x_3 = 0$

Find an orthonormal basis of the plane $x_1 - 5x_2 - x_3 = 0$ I'm having trouble with this problem. So I picked the vectors $u_1 = \begin{bmatrix}1\\0\\1\end{bmatrix}$ and $u_2 = \begin{bmatrix}5\\...
1
vote
0answers
40 views

Polar coordinate in Cartesian

My book states one can write polar coordinates $(\hat{r}, \hat{\theta})$as $$\hat{r} = \cos \theta i + \sin \theta j$$ $$\hat{\theta} = -\sin \theta i + \cos\theta j$$ Can someone explain how $$\hat{...
0
votes
1answer
19 views

Is there a unique projection map in this case?

Let $X$ be a Banach space over $\mathbb{C}$. Let $A,B$ be closed subspaces of $X$ such that $X=A\oplus B$. Assume that $||a+b||=||a||+||b||$ for each $(a,b)\in A\times B$. Then, does there exist a ...
2
votes
2answers
407 views

Why is the reduced echelon form of a set of independent vectors, the identity matrix?

If a matrix has linearly independent rows, then its reduced echelon form is the identity matrix. I haven't found a concise explanation for this... I have the whole notion in my head but I cannot ...
6
votes
4answers
507 views

What should “The Fundamental Theorem of Linear Algebra” assert? [closed]

Unlike some other basic fields of mathematics, linear algebra does not seem to have a universally agreed-upon fundamental theorem. This I imagine might be because the subject usually admits a lot of ...
0
votes
1answer
28 views

How to rotate a coordinate system in $\mathbb{R}^3$ through an angle about an arbitrary axis passing through origin?

The question spurred in my mind when I was asked the following: Find the transformation matrix T that describes a rotation by $120^\circ$ about an axis from the origin through the point $(1,1,1)$....
1
vote
1answer
53 views

Rank of orthogonal projection to prep of null-space, right-singular matrix and identity matrix

Let $\mathbf{A}$ be $m \times n$ ($m < n$) complex matrix and its SVD be $\mathbf{A}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^H$. Then we obtain an idempotent and Hermitian matrix, referred to ...
7
votes
5answers
3k views

Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?

I know if $A^TA=I$, $A$ is an orthogonal matrix. Orthogonal matrices also contain two different types: if $\det A=1$, $A$ is a rotation matrix; if $\det A=-1$, $A$ is a reflection matrix. My question ...