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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How can we find the inverse matrix?

How can we find the inverse of the matrix $$K=\begin{pmatrix}-(x+y) & y & 0 \\ y & -(y+z+w) & w\\ 0 & w & -(w+f)\end{pmatrix}$$ ??
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6 views

Why does complex conjugation permute the rows (columns) of a character table

If $\chi$ is the character of $\rho$, then $\overline{\chi}$ is the character of $\rho^*$ (dual) and $\chi_{irreducible} \iff \overline{\chi_{irreducible}}$. This implies complex conjugation ...
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2answers
11 views

Finding basis made of uninvertible matrices

Let there be transformation $T: \mathbb R_3[X] \rightarrow M_{2 \times 2}(\mathbb R)$, $T(ax^3+bx^2+cx+d)=\left[ \begin{matrix} a+d & b-2c \\ a+b-2c+d & 2c-b \\ ...
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0answers
6 views

determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
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0answers
14 views

Dimension of a linear space of polynomials

Let $P$ be a linear space of polynomials $P(x,y)$ of degree $\le2013$. Its subspace $V$ is formed by those polynomials for which the line integral $\oint_{x^2+y^2=R^2}f(x,y)ds=0$ for all R. What is ...
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1answer
18 views

Equations of the same plane

Are \begin{equation*} -x-4y+3z=-9~\text{and}~x+4y-3z=6 \end{equation*} equations of the same plane? I graphed them and they look the same, but I am not sure. Thanks
2
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1answer
27 views

Inequality in matrix norm

Let $\|\cdot\|$ be matrix norm on $M_n$.Why does $\|A\|_2 \le \|A\|^{\frac{1}{2}} \|A^*\|^{\frac{1}{2}}$? ($\|A\|_2 = \displaystyle\max_{\|x\|_2 = 1} \|Ax\|_2$)
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23 views

Taking Analysis I, Abstract Algebra I, and Theoretical Linear Algebra [on hold]

S.E advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the ...
3
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1answer
23 views

Invertible matrix over a ring and its eigenvalues

Eigenvalues and invertible matrices for fields and vector spaces: Let $K$ be a field (so $K^n$ is a $K$-vector space) and let $A \in K^{n\times n}$ be an $n\times n$-matrix. Then we have the following ...
3
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3answers
19 views

How to prove that this matrix is positive definite?

Let $\mathbf{A}=\begin{pmatrix}a^2+b^2 & b^2 & b^2 & ... & b^2 \\ b^2 & a^2+b^2 & b^2 & ... & b^2\\ \vdots & b^2 & \ddots & & b^2 \\ b^2 & \dots ...
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0answers
18 views

simplification of a linear algebra equation

I have been trying to simplify this equation, but with no success at all: So far what I have done is the following: but I get stuck in the last part, what am I missing?
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2answers
27 views

Commutative property of matrix multiplication (or lack thereof)

Assuming $A$ and $B$ are invertible matrices and are of proper dimensions to be multiplied (say, $2\times2$), is the following expression correct for all examples of matrices $A$ and $B$? ...
5
votes
1answer
39 views

Why does ${\lambda _i}(A) \ge {\lambda _i}(B)$?

Let $A,B \in {M_n}$ are Hermitian and $A-B$ has only nonnegative eigenvalues.Why does ${\lambda _i}(A) \ge {\lambda _i}(B)$ (for $i=1,2,\ldots,n$) ?
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0answers
15 views

Subrepresentations of $\mathbb{I} \oplus \xi$

$G=C_2=\{e,h \}$. $\mathbb{I}$ is the trivial representation and $\xi$ is the sign representation. Let us consider $\mathbb{I} \oplus \xi$ where $e \mapsto \begin{pmatrix} 1 & 0\\ 0 & 1 ...
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0answers
8 views

Euler Angle Transformation from righthanded to lefthanded cartesian coordinate system

I have a righthanded and a lefthanded cartesian coordinate system defined as follows: I have Euler angles (x, y, z) defined in the righthanded system and want to transform them to the lefthanded ...
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0answers
16 views

Why can commuting matrices be simultaneously upper-triangularized? [duplicate]

Say $A_i (i\in I)$ are commuting matrices in $\mathbb{C}^{n\times n}$. Show that there exists $U$ such that $U^*A_iU$ are upper triangular for all $i\in I$.
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writing a state of a dynamical system

Is it un/common to write the state of a dynamical system in the following manner: $$ \begin{pmatrix}x_{t+1} \\ v_{t+1} \end{pmatrix} = \begin{pmatrix}A & B \\ C & D ...
2
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1answer
18 views

Prob. 14, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hermitian positive semi-definite form

Let $X$ be a complex vector space, and let the map $h \colon X \times X \to \mathbb{C}$ satisfy the following conditions: For all $x, y, z \in X$ and $\alpha \in \mathbb{C}$, (i) $h(x+y, z) = ...
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0answers
10 views

Dual Space of an Euclidian Space is also Euclidic with a specific bilinear form

Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form, and let $\overline{\gamma}$ be defined by: $$\overline{\gamma}: V^* \times V^* \to K, \gamma(x, y) = ...
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0answers
21 views

A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically.

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of ...
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0answers
14 views

Inversion of the Burrows Wheelers Transform

The "Burrows-Wheeler Transform" in signal processing is a transformation which is used in for instance data compression and pattern recognition. It can be described in mathematical terms as: Start ...
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3answers
95 views

A question on linear algebra [on hold]

Let $V$ be a $n$-dimensional vector space and $T$ be a linear operator on $V$. Condition 1: there exists $0\neq v\in V$ such that $v, Tv,\ldots, T^{n-1}v$ are linearly independent. Condition 2: ...
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1answer
23 views

Span - linear algebra

I'm having some trouble in solving some exercises related to vector spaces, and I can't even start the solution. I need to check if the sets given span the same subset of the vector space $V$: (i) ...
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5answers
87 views

Proving that $\cos(2\pi/n)$ is algebraic

I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ ...
1
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2answers
27 views

Do $J$, the all-ones matrix of even order, always have eigenvectors consisting of entries $-1, 1$ only?

Do $J$, the all-ones matrix of even order, always have eigenvectors consisting of entries $-1, 1$ only? It seems so, vector having all its entries $1$ is one eigenvector for larest eigenvalue $n$ ...
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1answer
34 views

Definitions of $\mathrm{Hom}(V,W)$

I have the definition of a homomorphism as map such that $\varphi(g_1g_2)=\varphi(g_1)\varphi(g_2)$ I have the definition of $\mathrm{Hom}(V,W)$ as $$\begin{align}\mathrm{Hom}(V,W) &= ...
2
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0answers
27 views

A question on a matrix identity

Sorry for the not very specific title. I was hoping I could get some help with a result I do not understand. The following is from a book I am reading. What I do not understand is how from 9.9.6 one ...
1
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1answer
26 views

Eigenvalue for a conjugate operator.

$\newcommand{\lbrac}[1]{\left( #1 \right)}$ Let $V$ be a complex inner product space, and let $T:V\to V$ be a linear operator over $V$ and $T^*$ its adjoint. Suppose $\lambda$ is an eigenvalue of $T$. ...
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1answer
44 views

A statement regarding vector spaces…

Let $L$ be a vector space, and $U,W,V$ subspaces of $L$. Show: $$U\cap W\subseteq V \iff (U+V)\cap (W+V) =V$$ I've tried the following: Suppose that $(U+V)\cap (W+V) =V$. Since $0_L\in V$, we ...
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0answers
52 views

What exactly is antieigenvalue analysis?

I found a book in the library about antieigenvalue analysis and it is possibly the most unreadable piece of literature I have ever made an effort to understand. Unfortunately, every other resource I ...
1
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1answer
13 views

linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\ldots+c_mA_m$ is invertible. How to prove that for ...
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1answer
23 views

Solving 【(x^2+3x+1)^2】 by using a formula [on hold]

I know that (a+b)^2= a^2+2ab+b^2. Is there any formula to solve 【(x^2+3x+1)^2】?
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2answers
10 views

Solution set left unchanged after matrix multiplication?

If I solve $Tx=0$ where $T$ is some square matrix then if I multiply both sides by $T$ and solve for $T^2x=0$, will my x be the same? In other words if I were to multiply to both sides of the equation ...
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2answers
29 views

Orthogonal subspace of an orthogonal subspace

Let $V$ be an inner product space over $\mathbb{F}(\mathbb{C}\ or\ \mathbb{R})$, and let $W$ be a subspace of $V$. Assuming $V$ is finite-dimensional, I have proved that $(W^{\perp})^{\perp} = W$ ...
1
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1answer
46 views

Why is the following number always positive?

Consider two points in the Euclidean plane: $A=(A_1,A_2),B=(B_1,B_2)\in\mathbb{R}^2$, and some fixed real number $\lambda\in(0,1)$. The claim is that the following expression is always a positive ...
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1answer
27 views

The inverse of a matrix in which the sum of each row is $1$

Let $A$ be an invertible 10x10 matrix with real entries such that the sum of each row is $1$. Then choose the correct option. The sum of the entries of each row of the inverse of $A$ is $1$. The sum ...
4
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2answers
27 views

An invertible sparse matrix?

I'm not entirely certain about how to tackle this problem.... I hope you ladies and gents can help :) If $M\in M_{n\times n}(\mathbb{R})$ be such that every row has precisely tow non-zero entries, ...
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0answers
34 views

Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$.

Let $F$ be a field and let $n$ be a positive integer. Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F)$. (Exercise 438 from ...
2
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1answer
54 views

Why multiplying those 2 quaternions doesn't give the expected result?

Using a left-handed coordinate system, let Q = axisAngle({0,0,1}, 1/4*pi) * axisAngle({0,1,0}, 1/4*pi) be the quaternion representing the rotation "1/8 circle ...
1
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1answer
29 views

Show that any nontrivial ideal that is also a subspace of $M_{n\times n}$ over a field $F$ is maximal. [on hold]

I just need to show that if $A$ is a nonzero square matrix, then I can "build" the identity matrix $I$, Then the ideal must be maximal.
0
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1answer
40 views

Linear Transformations in Linear Algebra

We are given: Show how to evaluate a linear transformation for a specific vector $x$ , when the transformation is defined in the form $$T(x) = y$$ We know that a linear transformation is defined as ...
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functions (on intervals) in vector spaces [on hold]

I'm studying mathematical methods for solving physics and engineering problems. I've looked in a few books and I'm curious about functions being manipulated like vectors. Question: What should I ...
0
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1answer
31 views

Solving System of Equations modulo a prime

Consider the equation: $$ C \equiv HMH^{-1} \pmod{p}, $$ where $C,M, H$ are, say, $2\times 2$ matrices, and $p$ is an odd prime. The elements of the matrices $C, M$ are integers. The elements ...
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0answers
37 views

Prove a matrix is non-negative. [on hold]

Let $\textbf{r}_1$ and $\textbf{r}_2$ be $n \times n$ symmetric, diagonally dominant, Metzler matrix with eigenvalue $\max(|\lambda_i|)<1$ for both $\textbf{r}_1$ and $\textbf{r}_2$. Let ...
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0answers
10 views

Comparison between interpolation and Tikhonov regularization.

Interpolation is defined as finding a value of a function between two points and one can think of Tikhonov regularization as to estimate a suitable function under certain condition. Can we think ...
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1answer
24 views

Eigenvectors times diagonal matrix, still eigenvectors?

Suppose we have a $n\times n$ real symmetric positive definite matrix $\Sigma$, and $V=(v_1,...,v_n)$ whose columns are the eigenvectors corresponding to the $n$ eigenvalues $\lambda_1\geq \lambda_2 ...
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1answer
22 views

QR and Cholesky decomposition

A while ago I asked for help to develop a polynomial regression model using least squares, where the system was solved by cholesky decomposition, you can check it here Cholesky Polynomial Regression ...
2
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1answer
27 views

Singular matrix with entries in a ring. [duplicate]

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
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0answers
19 views

Isomorphism among quotient algebras [on hold]

Let we have two different quotient algebras $ A/B$ and $M/N$ such that $B$ and $N$ are nilpotent ideals of $ A$ and $M$. I want to define an isomorphism $ \phi: A/B \to M/N$.Can you help me how I ...
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1answer
38 views

Need a formula / method to get a value between 0 and 1 if a point lies in an area between two rectangles

I'm trying to figure out a way to construct a formula or method for the following process: If Point E is inside the smaller rectangle (red area), the value should be 1. If Point E is outside the ...