Questions tagged [linear-algebra]
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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.
84,610 questions
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What Does a Set Represent?
In Linear Algebra Done Right, Axler says that order and repetiton don't matter in a set. For example, the set {4} = {4,4} = {4,4,4} or {1,3} = {3,1}. My question is then what do these sets mean. What ...
2
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0answers
13 views
Householder matrices that make $H_n \cdots H_1 A = \begin{bmatrix} L \\ O \end{bmatrix}$
Problem
I have asked a similar question, but this time in a more general scenario.
Find Householder matrices $H_n, \cdots, H_1$ such that
$$H_n \cdots H_1 A = \begin{bmatrix} L \\ O \end{...
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1answer
20 views
Relationship of matrices with geometry
I have recently started learning linear algebra. Although I have studied in high school but that was just solving equations without knowing the purpose.
Coming straight to my question, for the ...
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0answers
20 views
Upper bound of infinity norm of stochastic matrix
If $W$ is a doubly stochastic, symmetric positive definite matrix, each entry of $W $ is nonnegative, then we know $\mathbf{1}$ is an eigenvector of $W$ corresponding to eigenvalue 1. If a vector $y$ ...
1
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1answer
22 views
The spectral decomposition of skew symmetric matrix
I have been studying the spectral decomposition of the matrices and figured out that it works for symmetric matrices but it wont work for the skew symmetric ones well, the sign of the final matrix is ...
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2answers
23 views
Bases for spaces of solutions
I have these spaces of solutions and im supposed to find an orthogonal basis for them. I can do this,if i have the regular basis for them. And i have no idea how to compute them:
a) $\begin{cases} 2x+...
1
vote
1answer
19 views
Finding every Householder reflection that converts $x$ to $y$, which are unit vectors.
Problem
Find every Householder reflection $H_v$ (with respect to $v$), such that
$$
y = H_vx
$$
where $x,y$ are unit vectors, i.e. $\Vert x \Vert_2 = \Vert y \Vert_2 = 1$, and $\langle x,...
2
votes
3answers
27 views
Find the values of a and b that make the equation system have infinite solutions
I'm given the following system of equations and I need to find the values for a and b so that the system has infinite solutions.
\begin{cases}
&3y &- &2z &= 5 \\
&-2x &+ &...
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0answers
14 views
transition matrix of dual basis
Q :Let $\left\lbrace v_i \right\rbrace^n_{i=1}$ and $\left\lbrace w_i \right\rbrace^n_{i=1}$ be basis of V and also let $\left\lbrace \phi_i \right\rbrace^n_{i=1}$ and $\left\lbrace \sigma_i \right\...
0
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1answer
26 views
Finding parameters a,b in a matrix with given eigenvalues
Im preparing for a linear algebra exam and im trying to solve the next exercise:
We are given a matrix A:
7 −4 0
a −7 b
3 −2 0
which eigenvalues are −1 and 1....
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0answers
11 views
KL divergence under exchange of arguments
Given probability vectors $A$ and $B$ which do not have nonzero entries (to avoid support related problems in the divergence), the KL divergence is
$$D(A||B) = \sum_i A_i(\log(A_i) - \log(B_i))$$
I ...
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2answers
20 views
How they commute $(T-\mu I)(T-\lambda I)^{p-1}(x)=^?(T-\lambda I)^{p-1}(T-\mu I)(x)$
Im currently reading "Linear Algebra, Fourth Edition,Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence"
At page 485 a theorem is located as follows:
Theorem 7.1
Let T be a linear ...
0
votes
1answer
18 views
Lowest Possible Average Correlation Between N Random Variables
Suppose $M$ is an $n \times n$ correlation matrix, with correlation $\rho_{i,j}$ between any pair of two random variables. What is the smallest possible average of the $\rho_{i,j}$ where $(i<j)$?
...
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0answers
16 views
Definition of positively oriented basis
Let $(U_\mathbb{R},A)$ be an oriented $m$-dimensional Euclidean space $(U_\mathbb{R},A)$, where $A$ is an alternating unimodular $m$-th order tensor, and subset $B=\{\vec{u}_1,\cdots,\vec{u}_m\}$ a ...
0
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1answer
19 views
Kronecker product on a matrix with structured blocks
I'm currently attempting to write a symmetric matrix with structured blocks into Kronecker-factorized form, but I'm not sure if the task is possible at all. My matrix takes the following form:
$$
M=
\...
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1answer
21 views
Eigenvalues and eigenvectors of $A_{1}$ and $ A_{2}=A_{1}^{T}$
We have a positive integer $n$ and two $n\times n$ matrices of real numbers, $A_{1}$ and $A_{2}$. For $j=1, 2$, we have the eigenvalues and eigenvectors $\lambda _{j}$ and $x_{j}$ of $A_{j}$. Show ...
0
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2answers
23 views
Linear Algebra Scalar and Vector Projection
I have a problem with one part of a question. I just need help with part b. How would I go about applying the equation to find the scalar and vector projection of v2 onto v1? Thanks!
1
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0answers
38 views
Prove the next proposition about f,g two endomorphisms that meet $f∘g=g∘f$…
Be f,g two endomorphisms $f,g∈End_k (E)$, where dim (E)=n>0 and finite, such that $f∘g=g∘f$. Prove that for all polynomial $p(x)$, $g(ker(p(f))) ⊆ ker(p(f))$.
What I tried to do is:
I want to prove $...
0
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0answers
17 views
How to minimize this energy by formulating it as a Poisson problem?
I have the energy function that I would like to minimize as:
$$\sum_{\{i, j\}}((h_i - h_j) - q_{ij})^2$$
This is applied over a 2D grid, where $q_{ij}$ is the relative height between two cells $i$ $j$,...
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2answers
24 views
How to find the number of x-length walks between two vertices of a triangle
This question is from my homework and I don't have any idea how to solve it. Find the number of 2019-length walks between two vertices of a triangle.
0
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0answers
26 views
Isomorphism and universal property
How can we verify that $\bigwedge^2\mathbb{R}^3\to\mathbb{R}^3$ is an isomorphism, by using the universal property of exterior power?
Thank you very much!
0
votes
1answer
21 views
What's the relationship between subspaces U and W in $U+W, U\cup W$, and $U \bigoplus W$?
Given two subspaces, $U$ and $W$ of $V$, I know that $U+W$ and $U\cup W$ are related in that $U+W$ is the smallest subspace containing $U\cup W$, but what's their relationship in $U+W, U\cup W$, and $...
0
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0answers
15 views
Example of an automorphism associated to a matrix A [on hold]
I know that a function $f_A$ can be associated to a matrix A. What's an example where an automorphism is associated to matrix A? Also, what happens if we compute $P(f_a)=f^2_A-f_A^0$?
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0answers
10 views
How to understand root subspace in linear algebra
According to the definition of the root subspace,there exists an exponent k that$(\mathcal A-\lambda\epsilon)^k v=0 $,but how to confirm this exponent?If it is an n*n matrix,what happens when the ...
0
votes
1answer
21 views
What's the dimension of the affine subspace $S: x-y+1=0$?
I know that an affine subspace is an hyperplane iff $dim(S)=dim(\mathbb{A})-1$. I know that the affine space in this case is $\mathbb{A}_\mathbb{R}^2$, I have to find the dimension of the affine ...
0
votes
0answers
8 views
Modification of Shannon wavelets for compactly supported maximally “square” wavelets?
The Shannon wavelets are maximally square in the sense that they are ideal band-pass filters in the Fourier domain.
Their amplitudes are literally box functions on frequency line and time shifts as we ...
1
vote
1answer
22 views
Moore-Penrose pseudoinverse and multiplication by diagonal matrix
Let $A \in \mathbb{R}^{n \times p}$, let $D$ be a diagonal matrix with positive entries. $\dagger$ denotes the Moore-Penrose pseudoinverse.
Is it true in general that:
$$(A^\top D A)^\dagger A^\top D ...
0
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0answers
13 views
Expectation Maximization (EM) for mixture of discrete distributions.
I am trying to get each components distribution from a mixture of discrete distribution but the model get stuck in m-step after just 1 or 2 iteration due to singular matrix error. I tried many ...
1
vote
2answers
17 views
related Theorem of linear functional
Q :Let V be a finite-dimensional vector space over a field k and let $V^\ast=Hom(V,k)$ be the dual space of V. Let $\left\lbrace v^i \right\rbrace^n_{i=1} $ be the dual basis of $V^\ast$. Then prove ...
0
votes
1answer
28 views
On inequality of exponentially activated, unit transform of vector
Let $x$ be a vector and $A$ a matrix. Let
$$
y := \dfrac{Ax}{\lVert A \rVert}
$$
bet a unit transform of vector $x$,
and
$$
\widehat{y}:= y/\lVert x\rVert
$$ the same transformation with ...
2
votes
4answers
60 views
Linear dependence of 3 vectors in $\mathbb{R}^4$
Let $a,b,c \in \mathbb{R},$ $\vec{v_1}=\begin{pmatrix}1\\4\\1\\-2 \end{pmatrix},$ $\vec{v_2}=\begin{pmatrix}-1\\a\\b\\2 \end{pmatrix},$ and $\vec{v_1}=\begin{pmatrix}1\\1\\1\\c \end{pmatrix}.$ What ...
1
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0answers
26 views
How to determine if these vectors are in the row space of this large matrix
A and B are 8 x 9 matrices. (i, j) entry of B is $i * j - 1$.
(i, j) entry of A is 0 if $i + j$ is divisible by 5 and is the (i, j) entry of B for the remaining entries.
I've managed to work out ...
1
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0answers
34 views
Characteristic polynomial and eigenvector of Frobenius matrix
Consider the following $n \times n$ matrix (I believe this is similar to companion matrix):
$$ A = \begin{pmatrix}
0 & -1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 0 & -...
0
votes
0answers
35 views
prove the rank of $AB$ [on hold]
1. $A$ is $m$ by $n$ matrix and $B$ is $n$ by $p$ matrix. Prove that $AB$ can be written as a sum of n matrices of rank at most one.
$A$ is $m$ by $n$ matrix with rank $m$ and $B$ is $n$ by $p$ ...
2
votes
4answers
71 views
What does shape does $\bigl\{x\in{\mathbb R}^3\bigm| A\times x=B\bigr\}$ define?
If vectors $A$ and $B$ are fixed, then by definition of the cross product, I know that if $B$ is not perpendicular to $A$ then there will be no solutions.
So assume $B$ is perpendicular to $A$, then ...
0
votes
0answers
16 views
Matrix Reduction with Positions
For a matrix $M$ and a set of positions $P=\{(x_i,y_i) \ | \ 1 \leq i \leq n\}$, define $f(M,P)$ as the sum of elements of $M$ at $P$, i.e., $f(M,P)=\sum_{(x_i,y_i)\in P}M_{x_i,y_i}$. Now given $M$ ...
0
votes
0answers
18 views
Matrix perturbation and eigenvector
$Ae=\lambda e$
$e^Te=1$
$A$ is a real matrix. $\lambda, e$ are real.
$(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$
Neglecting small terms,
$\Delta Ae +A \Delta e=\Delta \lambda ...
1
vote
1answer
34 views
Does the following result hold? [duplicate]
Suppose that $A$ and $B$ are symmetric and non-negative matrices.
Let $\rho(A)$ denote the spectral radius of $A$ and $\rho(B)$ denote the spectral radius of $B$.
Does the following result hold?
$...
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0answers
10 views
Prove that $\langle\mathbf{A}, [\mathbf{C}; \mathbf{0}]\rangle \leq \delta$ equals with $\|\mathbf{A_r}\|_*\leq\delta$ [on hold]
Given an arbitrary matrix $\mathbf{A}\in R^{n\times n}$ and the basis matrix set $\mathbb{S}=\{\mathbf{C}\in R^{r\times n}: \mathbf{C}\mathbf{C}^T=\mathbf{I}\}$.
$[\mathbf{C}; \mathbf{0}]\in R^{n\...
1
vote
1answer
15 views
Finding matrix from plane in kernal
Give an example of a matrix A such that $\ker(A)$ is the plane $2x − y + 3z = 0$.
I am not sure where to start, as I know that the $\ker(A)$ is the matrix of the plane, but I don't know how to go ...
0
votes
1answer
53 views
Why is the projection matrix $P = A(A^T A)^{-1} A^T$ left-multiplied by A?
Consider a vector space $V$ and its (orthogonal) subspaces $W$ and $U$. If $A$ is a matrix representing the linear map $T: V \rightarrow W$, and we want to project an element of $U$ onto $W$, why is ...
1
vote
1answer
26 views
How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?
How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$
here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
2
votes
2answers
50 views
Why is linearity not satisfied for this inner product?
The inner product given is $\langle u,v \rangle = u_{1}^{2}v_{1}^{2} + u_{2}^{2}v_{2}^{2}$. If I am thinking of linearity correctly, then all that means is the coefficients in front of each like term ...
0
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0answers
15 views
Rotation matrix from axis-angle representation
I came across the following representation of a rotation matrix given an axis of rotation $\begin{bmatrix}n_1 & n_2 & n_3\end{bmatrix}$ and an angle $\theta$.
I'm trying to figure out where ...
0
votes
2answers
34 views
Basis help: $v_1=[1;2;3]$
I'm not too sure why $[0;1;0]$ and $ [0;0;1]$ are a basis of $[1;2;3].$ Can anyone explain please?
From the textbook:
$v_1=[1;2;3].$
We find any basis for $\Bbb R^3$ containing $v_1.$
If we take ...
0
votes
1answer
44 views
Why are singular values of “complex” matrices always real and non-negative?
I've already read the following related questions on math.SE:
Why can't singular values be complex numbers?
Clarification on the SVD of a complex matrix
Why are singular values always non-negative?
...
0
votes
1answer
38 views
Find a real 3x3 matrix A that satisfies a quadratic equation
I want to find a real 3x3 matrix $A$ such that $A^{2}+A+I_{3}=0$ with the additional condition that it has at most one zero entry.
How can i compute his entries?
0
votes
1answer
15 views
Lattice in the Complex Plane
Let $\mathbb{Z}[i]$ be the ring of Gaussian integers, and $0\neq z\in \mathbb{C}$ be a complex number. Show that $\Lambda = \{zw:w\in\mathbb{Z}[i]\}$ is a lattice in the complex plane, find a basis ...
2
votes
3answers
42 views
Symmetric matrices containing $\{1, 2, 3, …\}$ in each line
I am a bit curious about an exercise.
I was supposed to prove that there is no matrix that has $1,2,3,4$ in each line and is symmetric. I did, by examination of all such matrices.
Now, there is a ...
0
votes
1answer
27 views
voltage matrix doesn't have a solution
Circuit Problem
Follow the link to the circuit that I need to solve for all the resistor voltage drops.
I have to make 5 linearly independent equations in order to solve for all 5 unknown voltages. ...
