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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$m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

As the title suggests, how do I see that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
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1answer
44 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
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votes
1answer
32 views

linear algebra (norm) [on hold]

Can someone explain to me the following definition - $\|T\|$ := $ \sup \{\|T(v)\| : v \in \mathbb{R}^n, \|v\| = 1\}$ where $T$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ and ...
-1
votes
0answers
26 views

How do you find the null space of an inconsistent system? [on hold]

For example, the augmented matrix: $$\left(\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$
5
votes
1answer
31 views

Finding an explicit eigenvector

Let $A$ be an $n\times n$ matrix over a field and let $\operatorname{adj}(A)$ denote its classical adjoint. Suppose all column sums of $A$ are zero so that $A$ is singular. If $\operatorname{rank}(A) ...
1
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2answers
51 views

Distinct eigenvalues and matrices problem

Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. It is given that if $v_1, . . . , v_n$ are eigenvectors for distinct eigenvalues $λ_1, . . . λ_n$ then $\{v_1, . . ...
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0answers
33 views

A question about minimizing the $\lambda_{max}$ over a set of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the ...
4
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1answer
43 views

Simultaneous orthogonal diagonalization of two matrices

Let $A=\begin{pmatrix} 1 & -2\\ -2 & 5 \end{pmatrix}$ and $B=\begin{pmatrix} -3 & 6\\ 6 & -10 \end{pmatrix}$. Obviously $A$ is positive-definite and thus we can simultaneously ...
1
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0answers
17 views

Action of the Symplectic Group on Siegel Upper Half Plane

Given $G= \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in Sp_{2n}(\mathbb{R})$ one can define an action on the symmetric $n \times n$ complex matrices with positive definite imaginary part by ...
4
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5answers
93 views

Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$

It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$ It is equivalent to the first order system in two variables $$\begin{bmatrix} x' \\ y' \end{bmatrix} = ...
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3answers
96 views

Good algebra book to cover these topics?

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your ...
2
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0answers
33 views

The Intution Behind Real Symmetric Matrices and Their Real Eigenvectors

I am wondering about the geometric intuition behind real symmetric matrices and their corresponding linear transformations. Is it possible to understand geometrically why real symmetric matrices ...
1
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1answer
24 views

Finding the minimal polynomial of a linear operator

Let $P=\begin{pmatrix} i & 2\\ -1 & -i \end{pmatrix}$ and $T_P\colon M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ a linear map defined by $T_P(X)=P^{-1}XP$. I need to find the ...
1
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1answer
66 views

Matrix exponential of $\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$

It is asked to evaluate the matrix exponential of $$A=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$$ It is not hard to do this, since this matrix have 3 ...
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0answers
32 views

Row rank$=$Column rank

This is one of the proofs given on Wikipedia. Let $A$ be an $m \times n$ matrix with entries in the real numbers whose row rank is $r$. Therefore, the dimension of the row space of $A$ is $r$. Let ...
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0answers
25 views

Distance/Similarity between matrices (different size) [on hold]

I have many matrices that have different size. Specifically, those matrices have the same number of rows but vary in the number of column. Each row is a different signal measurements, and each column ...
0
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0answers
11 views

Variable value estimation for given product/fracture values

I have a data set (time series) with given values for certain fractions xy = x/y (where x,y are not constant over time) Thus, there are following fractions: AB = A/B CB = C/B AD = A/D CD = C/D AE = ...
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1answer
11 views

Verifying expression with MP Pseudoinverse

Numerical simulations suggest that the expression $$ A=G^\dagger (PGG^\dagger P)^+G, $$ where $^+$ denotes the Moore-Pensore pseudoinverse, $P$ is the projector $$ P=I-\frac{1}{c^\dagger G^\dagger G ...
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0answers
30 views

Kernel of homomorphism

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I wanna show that the following homomorphisms $f_1$ and $f_2$ defined by $f_1: H\to GL_n(\mathbb{Z})$ $f_1(p)=P$ and $f_1(q)=Q$ ...
1
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1answer
31 views

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$.

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$. I'm not sure how to find these sets. I'm sure there is an elementary solution. Any solutions ...
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2answers
25 views

Row sum of $P^{m}$ when row sum of $P$ is $1$

Let $P$ be an $n\times n$ matrix whose row sum equals $1$. Then for any positive integer $m$ , what is the row sum of $P^{m}$ ? Now I took arbitrary $2\times 2$ matrix ...
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0answers
15 views

Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?
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0answers
13 views

Matrix & Linear Algebra - Rows Expressed as Linear Combinations of a Set of Linearly Independent Vectors

The question arises from a proof for showing that matrices and their transposes have the same rank, in the textbook Advanced Engineering Mathematics by Erwin Kreyszig. A matrix of a certain size and ...
0
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1answer
30 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And ...
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0answers
39 views

Retrieve the value of x,z and x [on hold]

I want to learn about HOW to calculation in order to retrieve the value of x, y and x. Do you have a recommended tutorial to for a beginner in relation to linear algebra in this specific case? I ...
1
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0answers
12 views

on triangular matrices and inverses [duplicate]

Suppose we have $A$, an upper triangular matrix. Can we conclude that $A^{-1}$ must be upper triangular as well? $A$ non singular. I mean it seems obvious but how can we prove it? by induction?
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1answer
44 views

The Calculation Process

I don't understand HOW the calculation is done to retrieve the value 9, -6 and 18? Thanks!
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0answers
30 views

Eigen vectors of a matrix multiplied with its transpose [on hold]

Do the eigen vectors of $A A^T$ and $AA^T$ belong to the row, column, null or left null spaces of the matrix $A$?
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1answer
41 views

Nicest operators on a vector space

Axler writes in his book that "nicest operators on $V$ are those for which there is an orthonormal basis of $V$ w.r.t which the operator has a diagonal matrix". i.e. orthonormal basis of $V$ ...
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0answers
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What is connected components of pseudospectra of matrix polynomial? . [on hold]

What is connected components of pseudospectra of matrix polynomial? Please see this link
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1answer
36 views

Sketching phase portrait of an ellipse

I have a system of linear ODE's as follows: $$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$ which has solution $$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}\cos2t\\-2\sin2t\end{bmatrix} + ...
0
votes
1answer
16 views

Assigning a specific value to components of a vector

So far, I've run into this twice and I'm not exactly sure how to make this connection myself, but in this case, I've been asked to find the dot product of $(i+j+k) \cdot (3i+2j-5k)$ I understand ...
0
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0answers
31 views

How to prove this decomposition

There are two vectors l=$(l_1,l_2)^T$, m=$(m_1,m_2)^T$, and a symmetric matrix S=$\begin{bmatrix}s_{11}&s_{12}\\s_{12}&s_{22}\end{bmatrix}$. Then, ...
2
votes
3answers
96 views

Proving a Set is a Vector Space [on hold]

for each $\left(x_1,y_1\right)\,,\, \left(x_2,y_2\right)$ that is an element of $\mathbb R$x$\mathbb R$ define $$(x_1, y_1) + (x_2, y_2) = (x_1 +x_2 + 2 , y_1 +y_2 -3)$$ And, for each $(x,y)$ that is ...
2
votes
1answer
39 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
2
votes
1answer
24 views

Using a linear function as a routine to determine a matrix

Let $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a linear function, i.e., $$F(\alpha x + \beta y) = \alpha F(x) + \beta F(y)$$ Suppose you are given a routine that returns $F(x)$ given any ...
-3
votes
3answers
47 views

A jazz concert brought in 128,000 on the sale of 8,100 tickets. If the tickets sold for $10 and $20 each, how many of each type ticket were sold? [on hold]

I am currently struggling on how to figure this out. I got as far as 165,000-81,000=84000. I am unsure what to do next. Thank you in advance!
1
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1answer
27 views

The relationship between diagonal entries and eigenvalues of a diagonalizable matrix

Let $\mathbf{C}$ be an $n\times n$ Hermitian matrix. Let $\dagger$ indicate a matrix conjugate-transpose. Let $\mathbf{V}\mathbf{D}\mathbf{V}^\dagger$ be the eigendecomposition of $\mathbf{C}$, where ...
4
votes
1answer
23 views

Diameter of unitary group.

Define a function$$N: \text{End}_\mathbb{C} \to \mathbb{R}_{\ge 0},\text{ }N(a) := \max_{\{v \in V\,:\, |v| = 1\}} |av|.$$ What is $$\max_{a, b \in U(V)} N(a - b),$$the "diameter" of the group ...
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0answers
16 views

Constrained optimization with several equality constraints

In maximizing a function of $n$ variables with $m$ equality constraints, it is required that the Jacobian derivative of constraints has full rank at optimal points. Can some one provide me with the ...
0
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1answer
45 views

Linear endomorphisms of $k(t)$

Let $k$ be a field and let $k(t)$ denote the field of rational variables in $t$. Is it possible to characterize all $k$-linear transformations from $k(t)$ to $k(t)$? Is $End_{k}(k(t)) \cong k(t)$ ?
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2answers
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matrix with all rows positive

I am thinking about a problem in a different area than linear algebra, but I came across a matrix with sum of entries of all rows positive, i.e. a matrix $A$ such that $\sum_{j} A_{ij}>0$ for all ...
5
votes
1answer
41 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
3
votes
2answers
47 views

About two systems of linear equations defining the same linear subspace

Assume that we have in a linear space $X$ a linear subspace $V$ over $F$ defined system of linear equations : $$ f_1=0, \\ f_2=0, \\ \,\vdots \\ f_k=0, $$ where linear functionals $f_1,\ldots,f_k \in ...
0
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0answers
57 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is fixed $n-1$, $\beta$ is chosen uniformly at random from the field of $p$ elements. We evaluate $P$ at some $x_i$ values. So we get ...
1
vote
2answers
59 views

Can Two Different Polynomials Agree on an open interval? [duplicate]

Question: For a high degree polynomial $P_1$ , can we have another polynomial $P_2$ that is a part of $P_1$ (or they agree on open interval)? TBN: This question is partially answered in ...
2
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1answer
29 views

$U(\mathbb{C}^n)$, $SU(\mathbb{C}^n)$ connected subsets of $M_n(\mathbb{C})$?

As the title suggests, is $U(\mathbb{C}^n)$ a connected subset of $M_n(\mathbb{C})$? How about $SU(\mathbb{C}^n)$?
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1answer
30 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
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2answers
29 views

Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$? Any solutions or hints are ...
3
votes
1answer
38 views

Does multiplication by a positive definite matrix preserve eigenvalues?

Let $A$ be a positive definite matrix and let $B$ a matrix. Then, $AB$ is similar to $A^{\frac{1}{2}}BA^{-\frac{1}{2}}$, which is in turn similar to $B$, so I get that $AB$ and $B$ are similar. ...