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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Given an area, calculate the angle of a wedge out of an annulus between a square and a circle

If we have a shape similar to this picture: Where the square length is less or equal to the circle's diameter, then I believe the term for the blue area is the annulus. I was wondering if it is ...
-1
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0answers
17 views

Signature of a complex matrix

If $A\in M_n(\Bbb R)$, we define its signature as the triple $(s^+,s^-,s^0)$, where $s^+,s^-,s^0\in\Bbb Z_{\ge0}$ denotes respectively the number of eigenvalues $>0,<0,=0$. How can we define ...
6
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1answer
37 views

Is an infinite system of (linear) equations solvable if all finite subsystems are?

I was wondering about the following. Let $A$ be an abelian group, $a_i$ variables indexed with some arbitrary set $I$ and assume we have an infinite set $E$ of linear equations in finitely many ...
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0answers
23 views

How do the generators of a group act on rank-2 symmetric tensors?

A rank-2 tensor $M_{ij}$ transforms as $M_{ij} \rightarrow O_{ik} O_{jl} M_{kl}$, where $O$ is some element of $SO(n)$. We can always get a symmetric tensor from $M_{ij}$ through $M_{ij}^s =M_{ij} + ...
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10 views

How to determine the minimum number of basis functions thats linear superposition best reproduces a set of curves?

How to determine the minimum number of basis functions that's linear superposition best reproduces a set of arbitrary curves?
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15 views

A question in matrix polynomial.

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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0answers
18 views

Why do uniform, strong and weak convergence coincide for finite dimensional vector spaces?

For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to ...
2
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2answers
21 views

Matrix representation with non-standard bases.

In chaprer 2.2 of Fiedberg's Linear Algebra is wroten about matrix representation. But all examples are only with standard ordered bases. I made a task to understand it. Please, could you show me ...
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4answers
61 views

Given vector $\vec x = \left\{ x_i\right\}_{i=1}^n$ find an algebraic expression for $\vec y = \left\{ x^2_i\right\}_{i=1}^n$

Given vector $$\vec x = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},$$ How can we write out vector $$\vec y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} := \begin{bmatrix} x^2_1 \\ ...
2
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2answers
13 views

Closest point of parameterized curve has orthogonal position vector to tangent

Let $\alpha(t)$ be a parameterized curve which does not pass through the origin. If $\alpha(t_0)$ is a point of the trace of $\alpha$ closest to the origin and $\alpha'(t_0)\ne 0$, show that the ...
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3answers
102 views

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$.

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. Any ideas?
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0answers
23 views

Derivations of important algebras?

After knowing the importance of studying derivations of an algebras(Why we wonder to know all derivations of an algebra?, this problem naturally raised "what is the space of all derivations of ...
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3answers
33 views

Finding the kernel of a linear map

Our exercise is to find all solutions to the equation $Ax = 0$, among others for the following matrix $$A =\begin{pmatrix} 6 & 3 & -9 \\ 2 & 1 & -3 \\ -4 & -2 & 6 ...
2
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1answer
25 views

Perpendicularity in matrix space

Let $K$ and $Q$ be symmetric real matrices such that $K+Q$ is positive semidefinite ($\ge0$). My question is two questions: Does $KQ=0$ imply $K\ge0$ and $Q\ge0$? Does trace$(KQ)=0$ imply $K\ge0$ ...
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0answers
27 views

Find the adjoint of the Linear OperatorT [on hold]

Find the adjoint of the Linear Operator $T:\mathbb R^3 \rightarrow \mathbb R^3$ defined BY $T(x,y,z) = (x+2y,3x-4z,y)$
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1answer
11 views

The projection $EF=E$ imply $M_2\subset M_1$?

Suppose $F$ is a projection on $M_1$ along $N_1$, $E$ is a projection on $M_2$ along $N_2$, if $EF=E$, does that imply that $M_2\subset M_1$?
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0answers
6 views

Is the basis for $\mathbb{R}^n$ where the $\mathfrak{so}(n)$ Cartan elements are diagonal, necessarily complex?

This is a follow-up to this and this question. The elements of $\mathfrak{so}(n)$ are antisymmetric in the standard basis $(1, 0, 0), (0, 1, 0), (0, 0, 1)$. This means that we have no diagonal ...
2
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1answer
23 views

$U^TA_1V$ is a rank-one matrix?

To give a little bit of context, the question I am asking is related to SVD decomposition. More specifically, we are trying to prove that the best rank one approximation for $A_1$ is $\sigma_1 u_{1} ...
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0answers
17 views

Can we express any matrix as an outer product expansion?

Suppose $XY$ is an $m $ by $n$ matrix, where $X$ is a $m$ by $k$ matrix and $Y$ is a $k$ by $n$ matrix. $y_i$ are the columns of $Y$ and $x_i$ are the columns of $X$. How do we know that ...
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1answer
38 views

Quadratic forms and midpoints

The midpoint of the vectors $u$ and $v$ is $w=\frac{u+v}{2}$. In euclidean geometry, an alternative characteristic of midpoints is $|v-w|=|u-w|=\frac{1}{2}|u-v|$. I wonder if this generalizes to ...
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0answers
30 views

More on linear algebra vector subspaces

I am continuing on my journey of trying to understand vector subspaces. Question: Let $F(-\infty,\infty)$ be the set of all real-value functions defined at each x in the interval $(-\infty,\infty)$. ...
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40 views

What do accountant's learn? [on hold]

Since a high school student can find compound interest, calculate stock yields, etc. what does an accountant learn in college? Is it just busy work?
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2answers
24 views

Closed under scalar multiplication [on hold]

The subset of $\mathbb{R}^2$: $\{ (x,y)| y=\frac{7}{2}x\}$ is a subspace of $\mathbb{R}$. How can I prove that the subspace is nontrival ?
2
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1answer
49 views

Diagonalization and Commuting Matrices

Attempt: I have shown part (ii) and I have found the eigenvalues and eigenvectors for A, B respectively and shown they can be diagonalised. I need help with (iii) and (iv), for (iii) I can't show ...
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2answers
33 views

How to find $f$ for a symmetric bilinear form?

Let's say we have the symmetric matrix:$$A = \left(\begin{array}{cc} 1&2 \\ 2&0 \end{array}\right)$$ How do I find the symmetric bilinear form of this $A$?
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1answer
42 views

What is the fastest, most correct way to solve this simultaneous of two linears?

\begin{eqnarray*} (x+2)/5-((y+2)/4) &=& 2-(x/3) \\ (x+5)/4+((x-y)/5) &=& y+5 \end{eqnarray*} What is the fastest, most correct way to solve this simultaneous of two linears?
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6answers
81 views

Find $x$ and $y$ - Why is there no answer?

I need to find $x$ and $y$ from the following equations: \begin{eqnarray*} 7x-3y &=& 8 \\ 14x-6y &=& 21 \end{eqnarray*} I my book it says there's "no answer". Can someone explain to ...
2
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2answers
43 views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = ...
1
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1answer
14 views

minimal polynomial and invariant subspace

I'm trying to solve the following problem: Let $T$ be a linear transformation on a finite dimensional vector space $W$. Suppose the minimal polynomial of $T$ is $p=g_1g_2$, where $g_1, g_2$ are ...
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1answer
36 views

Inverse of the sum of the inverse of 2 non-invertible matrices

Given that the following square matrices are non-invertible: $\bf A$, $\bf B$, and (A+B) UPDATE: Assume $\bf (A+B)$ is invertible. and given that $\bf (A+I)$, $\bf (B+I)$, and $\bf ...
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3answers
68 views

Does every invertible complex matrix have an eigenvector?

Over $\mathbb{C}$ does every invertible matrix have at least one non-zero eigenvalue and an eigenvector? I'm generally confused about eigenvectors and eigenvalues. I understand that eigenvectors are ...
1
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1answer
21 views

Eigenvalues of Certain Symmetric Block Matrix

What can we say about the relation between the eigenvalues of the following block matrix with identity diagonal blocks, and the singular values of the off-diagonal blocks: \begin{equation} ...
1
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2answers
34 views

All the cases for Image and Kernel

here alpha is a real variable, and I need to find the kernel and image for all values for alpha. Attempt: I can't seem to figure out all the cases which I need to evaluate, as the last two columns ...
1
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3answers
47 views

What is the definition of a $\mathbb{F}_2$-linear function?

To clarify, the function is $f:\mathbb{F}_2^m\rightarrow \mathbb{F}_2$. So, does it mean linear in each variable, or perhaps that each monomial is of degree $\leq1$? I know that sometimes terms have ...
1
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1answer
18 views

Interlacing Theorem on Singular Values

Does the Cauchy's interlacing theorem hold for "singular values" of matrices too? I saw on this publication first Theorem that it does. It states that singular values of a matrix interlace the ...
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0answers
21 views

Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
2
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1answer
19 views

Why $(U+iI_n)^2 = (U+iI_n)(U^*-iI_n)$?

Let $U$, a unitary operator and let $U+iI_n$, self-conjugate operator. Why is it true that: $$(U+iI_n)^2 = (U+iI_n)(U^*-iI_n)$$ We can evaluate both sides of the equation to get: $$(U+iI_n)^2 = U^2 ...
2
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2answers
37 views

If $UT=TU $, why is the range of $U $ invariant under $T $?

My Linear Algebra book says the following: Let $V$ be a vector space and $T$ be a transformation, which commutes with another transformation $U$. Then the kernel and range of $U$ are invariant ...
7
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1answer
112 views

Why we wonder to know all derivations of an algebra?

It is well-known that the space of all derivations of algebra of smooth functions on a manifold is its space of sections (vector fields on underlying manifold). But, I wonder to know why finding the ...
2
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1answer
49 views

$A,B$ are two real positive matrices then $\det (A+B) > \max(\det A , \det B)$

Let $A,B$ two square-real-positive matrices. Prove that $\det (A+B) > \max(\det A , \det B)$ So I found this solution: http://math.stackexchange.com/a/41478/160028 Basically, if $A=I_n$ and $B$ ...
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4answers
84 views

Is there always a matrix $X$ such that $X^2=A$?

Is it true that for every $A\in M_{2\times 2} (\mathbb{C})$ there's an $X\in M_{2\times 2} (\mathbb{C})$ such that $X^2=A$? For the matter of fact, I don't have a clue, other than evaluating the ...
0
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1answer
26 views

Proving that same solution set implies row equivalence

The question I am trying to solve is for a much simpler case: Suppose $R$ and $R'$ are $2\times3$ row-reduced echelon matrices and that the system $RX=0$ and $R'X=0$ have exactly the same ...
1
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2answers
42 views

Are those matrices congruent?

$$A = \left(\begin{array}{cccc} 1&0 \\ 0&-1 \end{array}\right), B=\left(\begin{array}{cccc} 1&0 \\ 0&2 \end{array}\right), C = \left(\begin{array}{cccc} 1&0 \\ 0&4 ...
2
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4answers
52 views

Vector Subspace

I have a question regarding vector subspaces: show $U=\{A\in M_{22} \mid A^2=A\}$ is not a subspace of $M_{22}$. I have said: let $A={(a_1, a_2, a_3, a_4) = \begin{pmatrix} a_1 & a_2 \\ a_3 & ...
0
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3answers
23 views

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$ I'm unsure how to proceed with proving this. Basically my idea is as ...
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1answer
19 views

Special Properties of Real Matrices With Real Distinct Eigenvalues

Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do ...
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0answers
10 views

Condition number of positive definite matrix after rectangular orthogonal transformation on both sides

What is a lower bound on the condition number of $B A B^{T}$ (besides the trivial $\operatorname{cond}(B A B^{T}) \ge 1$) where $A$ is an $n \times n$ symmetric positive definite matrix, $B$ is a ...
2
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2answers
38 views

Badly formed question? $\|x\|=1=\|y\|$ and $\|x+y\|=\|x\|+\|y\|$, there is a line segment in the unit sphere

Show that if a normed linear space $X$ contains linearly independent vectors $x$ and $y$ such that $\|x\|=1$ and $\|y\|=1$ with $\|x+y\|=\|x\|+\|y\|$, then there is a line segment contained in the ...
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1answer
40 views

What is $\frac{\partial x^TA^{-1}y}{\partial A}$?

I had trouble proving the following: If $ A\in \mathbb R^{n\times n}$ and $A$ is nonsingular, $x \in \mathbb R^{n\times 1}$, $y \in \mathbb R^{n\times 1}$, then $ \dfrac{\partial ...
0
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0answers
17 views

Does solution to particular system of O.D.E. with boundary conditions exist and unique?

I am dealing with the following system of $2$nd order differential equations: $$\ddot{y_k}+\frac{\ddot{y}_{N,k}}{2}+a\dot{y}_{N,k} = \gamma_k^2 y_k$$ where $y_{N,k} = \sum_{i=1,i\not=k}^{N} y_i ...