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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A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=End(V)$, defined as a ring of right operators on $V$. My questions are: (1) Is $V$ a simple right $E$-module? ...
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Generalization of Lines and Planes

Let $a_1,a_2,\dots,a_n$ be constants that are not all zero. An equation defines a line if and only if it can be written: $a_1x_1+a_2x_2+a_3=0$ An equation defines a plane if and only if it can be ...
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If all the columns of a linear system are pivot columns, then it has a unique solution.(Proof Verification))

I found the following theorem(CSRN) on the website http://linear.ups.edu/html/theorems.html . I am only interested in proving the r = n part. Theorem: Suppose A is the augmented matrix of a ...
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System of Equations, 3 Unknowns intersecting along a line parallel to the line where the other pairs intersect

I have this system of equations with 3 unknowns: Ax + By + Cz = D 5x + 3y + 2z = 4 -14x - 16y = 4 I need to find the values for A, B, C and D that will make it inconsistent and have each pair of ...
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Algebraic subspaces

How do I prove that $U=\{(x,y,z)|x\text{ is an integer}\}$ is not a subspace of $\mathbb{R}^3$? I understand that I have to show $U$ is closed or not closed under vector addition and scalar ...
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3answers
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The minimal poly of $T_w$ divides the minimal poly for $T$

I'm stuck on proving the following theorem. Let $W$ be an invariant subspace for $T$. The minimal polynomial for $T_W$ divides the minimal polynomial for $T$. We have \begin{equation} A ...
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Newton's method on a surface

I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - ...
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1answer
23 views

Is taking the real part required in vector orthogonality and projection?

In a real inner product space, two vectors are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$. Similarly, $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}) = ...
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1answer
21 views

Is it possible to find a companion matrix of a polynomial which is also hermitian?

The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial $$ a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ I can ...
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2answers
34 views

Does injective imply each $x$ matches to a unique $y$?

Injective means one-to-one matching, as in each $y$ is matched by only one $x$. However, does this mean that each $x$ matches only to one $y$?
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1answer
26 views

Computing eigenvalue of the adjacency matrix of a path

Let $A\in \{0,1\}^{n \times n}$ be the adjacency matrix of a path of length $n$, i.e. having ones on the two off-diagonals, and zeros elsewhere. How does one compute the eigenvalues of this? I know ...
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30 views

Create solid torus with geometric algebra

I have created an algorithm for tracking a vessel's centerline in 3 dimensions, and traveling through that centerline. My supervisor asked me whether I could add a small theoretical section on ...
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22 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
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1answer
56 views

Computing $\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr)$

Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ...
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4answers
247 views

How can I intuitively interpret this vector operation?

In reading through some very old source code that I inherited and came across a three-dimensional Euclidean vector operation that I can't seem to gain an intuition for. Transcribing the program code ...
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1answer
15 views

Determinant of a Certain 3 by 3 Block Matrix with Scaled Identity Blocks

What is the determinant or/and eigenvalues of the following 3 by 3 block matrix: $$\left[\begin{array}{ccc} \frac{3}{4}I & \frac{1}{4}I & \frac{1}{4}I \\ \frac{1}{4}I & \frac{3}{4}I & ...
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Eigenvalue of Block matrix: Adjacency of complete bipartite Graph

Let $A\in \{0,1\}^{mn \times mn}$ be the adjacency matrix of a complete bipartite graph with $m$ and $n$ vertices each, i.e. let $A$ be the matrix consisting of two blocks $A_1\in \{0,1\}^{m \times ...
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Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
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1answer
47 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
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1answer
43 views

How to Write a Vector Multiplication as a Trace of Matrix?

Let $\mathbf{w}_j\in\mathbb{C}^{M\times 1}$ and $\mathbf{h}_k\in\mathbb{C}^{M\times 1}$ be two complex vectors. How to prove this? ...
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1answer
60 views

Properties of matrices $M=UDU^*$ with $UU^*=Id$

I recently came across some matrices of the form $M=UDU^*$ (the superscript $*$ denotes the conjugate transpose), where $U \in \mathbb{C}^{r\times n}$ with $r<n$, $D \in \mathbb{C}^{n \times n}$ a ...
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1answer
37 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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1answer
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How to find the point in convex set $C$ that is closest to $y\notin C$?

How to find the point in convex set $C$ that is closest to $y\notin C$? $C=\{ x\in \mathbb{R^2}:(x_1-1)^2+(x_2-1)^2\le1 \}$ and let $y\notin C $ but $y\notin \mathbb{R^2} $.
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Point-Slope Equation of a line. Why is one answer incorrect and other is correct?

I am reviewing basic algebra. I am using quiz from the link this, and I solved the equation on paper and I get answer which is showing incorrect, I do not understand why is it wrong? It says my ...
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0answers
15 views

For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal? [duplicate]

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal? The condition number on $A$ is given as: $$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$ I tried ...
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2answers
37 views

Why is this a useful way to prove the characterisation of bases?

So I have come across the following theorem(see below)in my linear algebra notes and am slightly confused. I feel I understand both the statement and the proof, my confusion arises from the fact that ...
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40 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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1answer
26 views

inner product of positive semi definite symmetric matrices [on hold]

I have a positive semi definite symmetric matrix $X$, $(n\times n)$. let $X=vv^T$ s.t $\|v\|=1$. I came to a point where I am stuck to show which is: $v^TYv=\langle X,Y\rangle$ (How to show this ...
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Changing the subject of a formula

How do i make the subject of the formula $$A=\frac{m}{n}+\frac{n}{k}$$ into k? Also, does anyone know a website that solves this type of question for you? Mathway works fine, though I don't have ...
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2answers
35 views

Linear combination of basis function in logarithm space. Is it possible?

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
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1answer
30 views

What condition do I have set to have $x_j$ for $j=1,…,n$ to be non-negative?

I have $\sum_{j=1}^n q_{jj}(x_j-y_j)^2\le1$ What condition do I have set to have $x_j$ for $j=1,...,n$ to be non-negative? The book I am reading says $\sqrt{q_{jj}}\ge1/y_j$ but why? edit: ...
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1answer
69 views

Jordan canonical form of an upper triangular matrix

Find the Jordan canonical form of the matrix. Justify your answer. $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{bmatrix} $ My Try: The eigenvalues ...
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Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
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1answer
29 views

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$.

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$. My approach: Suppose span$\{u_1,\dots,u_n\}=W$, where$\{u_1,\dots,u_n\}$ is a set ...
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2answers
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$C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.

Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid ...
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No nontrivial invariant subspaces iff characteristic polynomial is irreducible

Say $V$ is a nonzero, finite dimensional vector space over $F$, and $T\in \mathcal{L}(V)$. I want to show that the only $T$-invariant subspaces of $V$ are trivial iff $f_T$, the characteristic ...
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Constructing a periodic function around $f(t): R \to R^3$

Definitions: $f(t): R \to R^3$ $\hat{\bigtriangledown}f(t) := \frac{df(t)}{dt} \frac{1}{|\frac{df(t)}{dt}|}$ $\vec{a}e^{\vec{b}x} := \vec{a}\cos(x) + \vec{b}\sin(x)$ $\vec{a}, \vec{b} \in H$ Is ...
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2answers
102 views

Looking for a gentle intro to Linear Algebra

Does anyone know of any gentle, introductory books to LA that assume little prerequisites, even in the way of vectors and matrices? I want something that will give intuition and reasonable proofs, and ...
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4answers
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why does the reduced row echelon form have the same null space as the original matrix?

What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?
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Basis of a product vector space

Let $E$ a vector space of dimension $p$ with $(e_1, \ldots, e_p)$ as a basis. Define the cartesian product vector space $F = v_1^\top E \times v_2^\top E \times \ldots \times v_n^\top E$ where the ...
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Project a signal $S(t) = \sum_0^{\infty}A(k)e^{if(k)t}$ to 3d domain $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $

Definitions: $\vec{v}e^{\vec{w}x} = \vec{v}\cos(x) + \vec{w}\sin(x)$ $ \psi_0(t) = x\hat{i}t + y\hat{j}t + z\hat{k}t, \ x,y,z\in R$ $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $ ...
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2answers
66 views

If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric?

If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?
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Can this be expressed in terms of linear constraints?

I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ ...
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Conditional expectation of a set of Gaussian variables

I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables $[x,y]$, the ...
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1answer
23 views

In what condition we have $(K^{-1})^\ast = (K^\ast)^{-1}$?

Suppose $X$ $Y$ are two finite dimensional Hilbert space. Assume $K$: $X\to Y$ is linear. My question is, in what condition of $K$ that $$(K^{-1})^\ast = (K^\ast)^{-1}?$$
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1answer
19 views

Geometric and Algebraic Multiplicity, zero dimensions

The eigenvalues are $\lambda =0$(because we have multiplication here), $\lambda =1$, and $\lambda =2$ for the given characteristic equation, and as (a) states, that $GM\le AM$. Now, I want to know ...
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1answer
27 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
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1answer
24 views

Different representations of a matrix in reduced row echelon form

EDIT: I decided to ask this question after working on this particular problem. I had the stupidity to think that row-reduced = row reduced echelon form. Brain fart, nothing more to see here... ...
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Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
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16 views

Maximum in a linear system of equations

I have a system of equations with a tridiagonal coefficient matrix: $$ \alpha_i f_{i-1} + f_{i} + \beta_i f_{i+1} = \Gamma_i $$ , where $i$ goes from 1 to $n$. For a given $M$, what constraints ...