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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1answer
14 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
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1answer
43 views

Rotation Matrix and programming [on hold]

I am actually programming in Android. An android tablet as a lot of sensors including one that gives the rotation vector of the tablet. (See ...
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0answers
28 views

Solving linear equation for low rank matrices

Consider $Ax=b$ where $A$ is invertible, so we have $x=A^{-1}b$. Now, let's consider a low-rank approximation of $A$, say $\bar{A}$ such that $rank(\bar{A})\leq r$ and $||A-\bar{A}||_F\leq \epsilon$ ...
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1answer
25 views

Matrix of Killing form a Lie algebra

Let $L$ be the Lie algebra with basis $B = \{u,v,w\}$, with $[u,v] = w, [v,w] = u, [w,u] = v$. Question : Find the matrix of the Killing form $\kappa$ of $L$ with respect to $B$. I have come across ...
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1answer
23 views

Relations between Kernel and image [on hold]

Let $T:V \to V$ be a linear transformation. Prove that $T^2=0$ if and only if $\operatorname{Im}(T) \subset \operatorname{Ker}(T)$.
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3answers
25 views

Linear transformations and conditions

Let $T: \mathbb{R}^n\to \mathbb{R}^n$ be a linear transformation given by \begin{equation*} T(x_1, x_2,...x_n)=(a_1x_1, a_2x_2,...,a_nx_n). \end{equation*} a) Under which conditions on ...
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1answer
40 views

Prove that the dual -space of the dual-space of V is isomorphic to V without using bases

Given a vector space $V$ the dual space $V^*$ is the space of all linear operators from $V$ to $\mathbb{C}$. $V^*$ is itself a vector space and I know how to prove $V \cong (V^*)^*$ by using a ...
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3answers
41 views

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$?

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$? My thinking is: $\langle u+v+w,v \rangle =0 \iff \langle u,v \rangle + \langle w,v \rangle = -1$ How do i ...
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2answers
38 views

eigenvalues of A - aI in terms of eigenvalues of A

I am stuck with this question of my assignment where given that A is nxn square matrix and a be a scalar it is asked to - Find the eigenvalues of A - aI in terms of eigenvalues of A. A and A - aI ...
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21 views

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$?

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$? Obviously $$N(z_1+z_2)=\left(\sum_{n=1}^{\infty} ...
2
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3answers
48 views

Least Squares method and Octave/Matlab [on hold]

I'll try to be as clear as possible so that you understand what I'm trying to do and can help me I have twelve pairs of data $(x_1,y_1),....,(x_{12},y_{12})$ and from this data we established a model ...
2
votes
2answers
59 views

Suppose $A^2B+BA^2=2ABA$.Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$.

Let $A, B \in M_n(\mathbb{C})$ be two $n \times n$ matrices such that $$A^2B+BA^2=2ABA$$ Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$. Here is the source of the problem. ...
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3answers
19 views

Trig algebra problems, taking out a factor of tan

$$ \sin\theta-\cos\theta=0 $$ ${\sin\theta\over\cos\theta}=\tan\theta $ $$ \cos\theta (\tan\theta-1)=0$$ $$\tan\theta=1$$ $$\cos\theta=0$$ $$\theta=45, 90$$ However the second solution is not true ...
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0answers
32 views

Necessary condition for existence of a positive solution of a linear system

I would like to know what are the necessary conditions of existence of a positive (componentwise) solution of the system : Ax=b, with A a square ...
2
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0answers
45 views

Is there a name or symbol for the matrix division resulting in a scalar?

I am not talking about the inverse matrix, $A^{-1}$ which gives $A\times A^{-1}=I$, but rather the operation $\frac{1}{n}tr(\space\cdot \times A^{-1})$, which gives 1 when applied to a $n\times n$ ...
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0answers
23 views

Let $H$ be a Hilbert Space with$\langle \cdot,\cdot \rangle$ and $E_1=\{w\in H : Pw=w\}$, show $E_1$ is closed.

Let $H$ be a Hilbert Space with $\langle \cdot,\cdot \rangle$ and $E_1=\{w\in H : Pw=w\}$ with $P:H\rightarrow H$ is linear, $P^2=P$ and $\langle Px,y \rangle=\langle x,Py \rangle \forall x,y\in H$. ...
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votes
2answers
69 views

Show that $\det(A) > 0$

Let $(a_{ij})$ be a real $n \times n$ matrix satisfying, $a_{ii} > 0 \space (1 \leq i \leq n) ,$ $a_{ij} \leq 0 \space (i \ne j, 1 \leq i,j \leq n) ,$ $\sum_{i=1}^ {i=n} \space ...
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0answers
18 views

Sum of abs of negative eigenvalues divided by sum of abs of all eigen values.If the result is convex?

Let $\lambda_1 (X)\geq \lambda_2 (X)\geq\ldots\geq\lambda_n (X)$ denote the eigenvalues of a matrix $X\in S^n$. Let $f(X)= \sum_{i\colon λ_i<0}|\lambda_i(X)|$ and $g(X)= \sum_i|\lambda_i (X)|$. ...
3
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0answers
17 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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2answers
49 views

What are the possible eigenvalues of matrix $A$ that satisfies $A^2=-I$? [on hold]

Let $A$ be a matrix such that $A^2=-I$, where $I$ is identity matrix. What are the possible eigenvalues of $A$?
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1answer
20 views

Coordinate vectors

Find the vector x $\in$ $\mathbb{R^3}$ whose coordinates with respect to the basis $$B = \{(-2,2,2),(3,-2,3),(2,-1,1)\}$$ are $[x]_B = [2;1;1].$
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2answers
58 views

problem about symmetric positive semi-definite matrix

Let $A,B$ be symmetric positive semi-definite matrix with real entries I have to show that $ Im(A) \subset Im(A+B)$ if $tr(AB)=0$ then $ AB=O $ I know that a symmetric matrix A is positive ...
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0answers
8 views

Get row height from content Length and column Width [on hold]

I got a problem computing the height of a row table using individual column width and ...
0
votes
1answer
35 views

cross product of vectors - distance from a line.

I have the following question : Let $A$ and $B$ be distinct points in $\mathbb R^3$. Show that the distance, d, of the point $P\in{\mathbb R^3}$, from the line through A and B is given by ...
0
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1answer
42 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
6
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3answers
218 views

Are a row vector and a column vector the same thing?

Suppose I have $$ A = \begin{bmatrix} a\\b\\c\\d\\ \end{bmatrix}$$ $$ B = \begin{bmatrix} a& b& c & d\\ \end{bmatrix}$$ Now, I know $A = B^T$. But in what sense are these different ...
4
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3answers
49 views

The dimension of centralizer of a Matrix.

Let $A$ be a $n\times n$ matrix with characteristic polynomial $$(x-c_{1})^{{d}_{1}}(x-c_{2})^{{d}_{2}}...(x-c_{k})^{{d}_{k}}$$ where $c_{1},c_{2},...,c_{k}$ are distinct. Let $V$ be the space of ...
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0answers
16 views

linear map as a product of certain linear maps

Why a linear map $T$ can be written as a composition of linear transformations of the following special forms: 1)$T(x_1,\dots,x_k)= (x_{\pi1},\dots,x_{\pi k})$, where $\pi$ is a permutation ...
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0answers
31 views

Linear Algebra : Vectors [on hold]

Please can someone shed some light on how I can tackle this question: consider the points $P(3,-1,4) Q(6,0,2)$ and $R(5,1,1)$. a) find the point $S$ in $ \mathbb R^3$ whose first components is $-1$ ...
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1answer
16 views

Composition and linear transformation

Prove that if $T: V \to V$ and $S:V \to V$ are non-null linear transformations such that $T \circ S=0$, then $T$ is not injective.
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17 views

Is there a relation between boundary value problems and ill-posed problem?

This question is related to the links: First defining ill-posed problems: http://www.encyclopediaofmath.org/index.php/Ill-posed_problems and second the Green's theorem which is related to the ...
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0answers
29 views

Vector Identities [on hold]

I am trying to simplify a vector equation, and I feel like this is something, but I am not sure what. Has anybody seen something like this before? $\|\mathbf{a}\|^2 ...
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1answer
30 views

System of linear equations: and a small perturbation

If $Ax=b$ and $Ax'=b'$ where $x'$ and $b'$ are $x$ and $b$ with a small perturbation, the following inequality will always hold: $ (\left\lVert x-x' \right\rVert/\left / \lVert x \right\rVert) \le ...
1
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1answer
34 views

Let $G$ be a graph on $n$ vertices, where $n \geq 3$. Suppose that $\Delta(G) \geq n/2$. Can $G$ have more than one component?

Let $G$ be a graph on $n$ vertices, where $n \geq 3$. Suppose that $\Delta(G) \geq n/2$. Can $G$ have more than one component? i did this for $n=3$ $\Delta(G) \geq 1.5$ as $\Delta.1$ component. for ...
4
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0answers
46 views

How to calculate this special determinant

$$\left| {\begin{array}{*{20}{c}} 1&{{a_1} + a_1^{ - 1}}& \cdots &{a_1^{n - 1} + a_1^{n - 3} + a_1^{n - 5} + \cdots + a_1^{1 - n}}\\ 1&{{a_2} + a_2^{ - 1}}& \cdots &{a_2^{n - ...
3
votes
0answers
61 views

Find a symmetric matrix B that makes ABC symmetric, A,C known

I have two known matricies $\bf{A} \in \mathbb{R}_{nxm} $, $\bf{C} \in \mathbb{R}_{mxn}$ with $m>n$. I'm trying to find a $\bf{B} \in \mathbb{R}_{mxm}$ that is symmetric and makes $\bf{ABC}$ ...
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0answers
21 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
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0answers
37 views

The length of a shortest path from $u$ to $v$ in a connected graph $G$ equals the level of $v$ in any BFS tree of $G$ with $u$ as root

I am studying graph theory but I cannot solve this question. Can you help me? "The length of a shortest path from $u$ to $v$ in a connected graph $G$ equals the level of $v$ in any BFS tree of $G$ ...
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0answers
12 views

Literature Reference for transformations through vector spaces

I am trying to understand the transformations through vector spaces: Problem 1. Let's say we have orthonormal basis $B=\{v_1, v_2, \ldots, v_n\}$ spanning the vector space $V$ and basis $B_1=\{u_1, ...
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1answer
19 views

Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
0
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1answer
35 views

Linear algebra simplification question

I am currently reading a paper that does the following simplification. I have broken it down and worked it out by some examples, but can anyone show me how they made this simplification using the ...
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0answers
24 views

Integer solutions to Equation System

Ax=b, A, b contain integer entries. If i apply integer row/column reduction (multiply row/column +-1, add integer multiple of one row to another, exchange rows) to both A and b, does the solution ...
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2answers
36 views

Isomorphisms between subspaces

Let $U$ and $W$ two subspaces of a vector space $V$. Let $T: U \times W \to V$ be a function, defined by $T(u, w) = u + w$. Show that i) $T$ is a linear transformation, ii) Image of $T$ ...
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21 views

Is it true that any 20 regular graph has a path of length at least 10? [duplicate]

hi i am confused with this question."Is it true that any 20 regular graph has a path of length at least 10?" Does this make sense?
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0answers
22 views

Gradient of a function at the boundary of a constant region

Seemingly an easy thing to do, I had difficulty to find an answer for the following: Let's assume we have a function $f(x)$ which is defined as $f:\mathbb{R^n} \to \mathbb{R}$. The function has ...
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1answer
39 views

Can a graph with no cut edges contain a cut vertex [on hold]

hi can you help with this question. "Can a graph with no cut edges contain a cut vertex?"
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1answer
13 views

Form matrix and calculate it's determinant

I need help with this problem: For every $i,j \in \{1,2,...,n\}$ is $d_{i,j}=min\{i,j\}$. Calculate determinant of a matrix $[d_{i,j}]_{n_Xn}$. Is it right that all the elements of this squared ...
1
vote
1answer
18 views

Prove that the identity matrix applies in 4 x 4 alternate matrix.

Let invertible matrix A size 4 x 4 and let matrix B be the matrix that is the result of alternating the first two lines of A and adding in the fourth line the double amount of the third line. Is B an ...
3
votes
1answer
28 views

When an eigen vector is zero vector

Question : Mike opens a bank account with an initial balance of 2000 dollars. Let b(t) be the balance in the account at time t. Thus b(0)=2000. The bank is paying interest at a continuous rate of 3% ...
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2answers
25 views

Characteristic Polynomial of $4×4$ matrix

Let $A\in M_{4}(\mathbb{F})$ where $a_{ij}=1, 1\leq i,j \leq 4$. Other than solving $det(xI-A)$, is there an easy way to get the characteristic polynomial of $A$? Is there a way to determine if $A$ ...