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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Problem From Golan's Linear Algebra Book-Linear Transformation

I am unable to approach the following problem from Golan's The Linear Algebra a Beginning Graduate Student Ought to Know. Let $V$ be a vector space finite dimensional over a field $F$, the ...
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2answers
28 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 $(W^{\perp})^{\perp} = W$ ...
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3answers
17 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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5answers
245 views

$T:\Bbb R^2 \to \Bbb R^2$, linear, diagonal with respect to any basis.

Is there a linear transformation from $\Bbb R^2$ to $\Bbb R^2$ which is represented by a diagonal matrix when written with respect to any fixed basis? If such linear transformation $T$ exists, then ...
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2answers
20 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 $B$? ...
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0answers
13 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
15 views

Taking Analysis I, Abstract Algebra I, and Theoretical Linear Algebra

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 ...
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0answers
7 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 ...
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0answers
7 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 ...
2
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0answers
15 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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0answers
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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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How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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1answer
16 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
27 views

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 ...
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1answer
323 views

Which of the subsets of $\mathbb{R^{3\times 3}}$ are subspaces of $\mathbb{R^{3\times 3}}$?

The invertible $3 \times 3$ matrices The $3\times 3$ matrices whose entries are all integers The $3\times 3$ matrices with all zeros in the third row The non-invertible $3\times 3$ matrices The ...
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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 ...
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0answers
50 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 ...
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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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3answers
93 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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0answers
19 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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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 $\cos\frac{2\pi}{n}$ ...
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0answers
12 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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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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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
22 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) ...
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
33 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) &= ...
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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
25 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 ...
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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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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 ...
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3answers
64 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that ...
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2answers
36 views

Let $A$ be a complex $2$ by $2$ matrix having distinct eigenvalues $a, b$. Show that $A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A - aI)$.

Let $A\in\mathscr{M}_{2\times 2}(\mathbb{C})$ be a matrix having distinct eigenvalues $a\neq b$. Show that, for all $n > 0$, \begin{equation*} A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - ...
3
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1answer
47 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. (Exercise 705 from Golan, The Linear Algebra a ...
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1answer
22 views

Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that ...
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2answers
54 views

Let $V$ be a vector space and $\alpha$ a nilpotent endomorfism (of degree $k$), how can I show that $\alpha(x)+x$ is epic?

Let $V$ be a vector space and $\alpha$ a nilpotent endomorfism (of degree $k$), how can I show that $\alpha(x)+x$ is epic? (Exercise 770 from Golan, The Linear Algebra a Beginning Graduate Student ...
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1answer
23 views

Let $\dim(V)<\infty$ . Do there exist endomorphisms of $V$ satisfying the condition that $σ_1 + αβ − βα$ is nilpotent?

Let $V$ be a vector space finitely-generated over $\mathbb{C}$. Do there exist endomorphisms $α$ and $β$ of $V$ satisfying the condition that $σ_1 + αβ − βα$ is nilpotent? (Exercise 771 from ...
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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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Equivalence class of matrices on linear form

We well know that if $M$ is a matrix on a field $k$ then the equivalence class of $M$ is uniquely determined by its rank (where $A \sim B$ if $\exists P,Q $ invertibles such that $PAQ^{-1}=B$). ...
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1answer
37 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 ...
0
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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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1answer
26 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 ...
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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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2answers
51 views

Do positive-definite matrices always have real eigen values?

Do positive-definite matrices always have real eigenvalues? I tried looking for examples of matrices without real eigenvalues (they would have even dimensions). But the examples I tend to see all ...
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 ...
4
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1answer
31 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
9
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7answers
881 views

What is the idea behind a projection operator? What does it do?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can anyone help me? I don't need examples or the definition - I want to know why and ...
2
votes
4answers
35 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
13
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4answers
143 views

Matrices such that $M^2+M^T=I_n$ are invertible

Let $M$ be an $n\times n$ real matrix such that $M^2+M^T=I_n$. Prove that $M$ is invertible Here is my progress: Playing with determinant: one has $\det(M^2)=\det(I_n-M^T)$ hence ...