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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Matrix $AB = 0$ , so $A$ and $B$ are not invertible

I am trying to show that if a matrix $AB = 0$ , then the matrices $A$ and $B$ are not invertible.
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14 views

Rank of zero determinant matrix

if we multiply two 1 column vectors, by taking transpose of one of them, and get an nxn matrix called A, det(A) is 0 right? Or am i mistaken? Also i know that rank of A should be less than n. But can ...
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1answer
5 views

Is upper Hessenberg form preserved through similarity transformation

Suppose $X$ is non-singular and $M$ is upper Hessenberg. Is $X^{-1}MX$ also upper Hessenberg.
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26 views

Number of solutions of an equation

Fix a vector $x\in\{0,1\}^n$, and let $a$ be a random vector in $\mathbb{Z}^n_q$ for some prime $q$. Consider $y=ax$, and $S=\{x'\mid ax'=y\}$. I want to compute the probability that $\lvert S ...
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0answers
7 views

What is an inner-outer iteration?

Inner-outer iterations are used in papers, for finding a stationary point of a system or in optimization. It is not clear, what is called an inner-outer loop though? Is it a nested loop where the ...
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0answers
22 views

How do I express, algebraically, this comparison of two sets of sets?

Say I have two sets (A and B) containing sets of the same integers. For example: $A_1 = \left\{{1,2}\right\}$, $A_2 = \left\{{3}\right\}$, $A_3 = \left\{{4,5,6}\right\}$ $B_1 = ...
2
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1answer
23 views

Showing that (on $\mathbb{R}^n$) the $\|\cdot\|_\infty$ norm is weaker than any other norm

Showing that (on $\mathbb{R}^n$) the $\|\cdot\|_\infty$ norm is weaker than any other norm. I am doing past papers and the question is this: "Prove that any norm on $\mathbb{R}^n$ is weaker than the ...
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4answers
62 views

Exercise about Matrix diagonalization

Well I have to diagonalize this matrix : $$ \begin{pmatrix} 5 & 0 & -1 \\ 1 & 4 & -1 \\ -1 & 0 & 5 \end{pmatrix} $$ I find the polynome witch is $P=-(\lambda-4)^2(\lambda-6)$ ...
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2answers
30 views

Proving rank of $AB$ is at most equal to rank of $B$

$A=m\times n$ matrix. $B = n\times p$ matrix. Prove that the rank of of the product $AB$ is at most equal to the rank of $B$. Current state of my work: (1) First idea: show that the kernel of $B$, ...
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2answers
39 views

Proof of a real eigenvalue

Let $A$ be a $2\times2$ matrix $A=\begin{pmatrix}a&b \\ c&d\end{pmatrix}$. I found the characteristic polynomial which is $T^2-(a+b)T+ad-bc$. It can be written as ...
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1answer
11 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. What are the conditions on $X$ such that $\{[X,g_i]\; \big| \; g_i ...
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0answers
6 views

Singularity of a positive linear combination of rank one matrices

Given a set of rank one matrices $A_1,..,A_n$, we need to find out if there exists $x \in \mathbb R^n$ with $x\gg 0$ (i.e, positive) such that $$ \sum_{i=1}^n x_i A_i = x_1 A_1 + .... + x_n A_n $$ ...
2
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1answer
12 views

Number of binary solutions given the rank of a matrix

Let $A\in\mathbb{Z}^{n\times n}$, and consider the system of equations $Ax=b$ s.t. $x\in \{0,1\}^n$. Assume $A$ is not full rank. Is it correct that the number of solutions is $2^{n-r}$ where $r$ is ...
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0answers
8 views

QR method for Hessenberg matrices

In trying to implement the method, my approach is to use a reduction to Hessenberg form, and then to iterate using a QR method of Givens rotations. However, I am having trouble successfully ...
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1answer
20 views

Simple Linear Algebra Question about consistency of a linear system

I just wanted to confirm. Can we say that a system of linear equation, which is consistent, must have infinitely many solutions if there are more number of equations than variables? I'm guessing ...
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1answer
21 views

Find the solution following [on hold]

What conditions must be placed on a , b , and c so that the following system of equations has a solution? $$x+2y-3z=a$$$$ 2x+6y-11z=b$$$$x-2y+7z=c $$ and find it if $a=1,b=2,c=1.$
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27 views

Action of matrix on symmetric products

Suppose that $M : V \to V$ is a linear map of a finite-dimensional vector space. This induces a linear map $M_n : \operatorname{Sym}^n(V) \to \operatorname{Sym}^n(V)$ for any $n \geq 1$. Is there a ...
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0answers
8 views

Parametrization of the split orthogonal group O(n,n)

I would like to find or construct an explicit parametrization of the $2m$-by-$2m$ matrix representation of the real indefinite orthogonal group $O(m,m)$ associated to the bilinear form with matrix ...
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2answers
23 views

Inner Product Space and Linear Mapping Theorem

I'm having some trouble proving the following theorem: Let $($$X$,$\langle\cdot | \cdot\rangle$$)$ be an inner product space and $f: X \to \mathbb{R}$ a linear mapping. Prove that there exists a ...
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1answer
16 views

Which type of correlation should I use?

I am beginner in statistics. I have excel table with few columns. I would like to find correlation between the variables. I have to make an essay to my boss and he wants concrete answers. I searchin ...
3
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2answers
55 views

(Linear algebra) if $A$ is normal matrix then, eigenvectors of $ A$ are orthogonal.

I know that the eigenvectors of a unitary matrix are orthogonal. Then is that also true for a normal matrix? How do I prove?
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3answers
33 views

If $V$ is a vector space $\neq$ the vector space of its additive identity alone, must $V$ have a subspace $\neq V$?

It seems to me highly plausible that every vector space $V$ such that $V$ does not consist of the additive identity alone has a (nontrivial) subspace $\neq V$. But I have not yet seen a way to prove ...
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1answer
12 views

Is decomposition of a semisimple Lie algebra unique?

A semisimple Lie algebra is defined to be the sum of simple Lie algebras. But is this decomposition to simple Lie algebras unique? If not can you give an example?
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2answers
15 views

Does the definition of eigenvalues work for non-injective linear maps?

I have this result that says: Let $T:V\to V$ be a linear transformation and let $I$ be the Identity map ($I(v)=v$), then $\alpha$ is an eigenvalue of $T$ iff $T-\alpha I$ is not one to one. My ...
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1answer
22 views

Is the sum of rank one matrices positive semi definite?

Consider $n$ vectors $x_i$ such that $x_i\in\Re^n,i=1,\dots,n$, the matrix: $$ A=\sum_{i=1}^n x_i x_i^T, $$ is positive semidefinite? Are there any results in literature?
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1answer
19 views

Proving if a transformation is linear.

I have a terrible understanding of Linear Algebra so I'm trying every resource out there that I can. Hopefully my questions won't come off as idiotic. Anyway, we have a transformation from $R^2$ to ...
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1answer
8 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
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1answer
26 views

Lipschitz constant of L2 difference

What is the Lipschitz constant of $$f(A)=||Ax||_2-||Ay||_2?$$ In particular, is it $||x-y||_2$, i.e. is it true that given $A,B,x,y$, the following inequality holds: $$|f(A)-f(B)|\leq ||x-y||_2 ...
1
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1answer
31 views

Convergence of the LR algorithm for $2\times 2$ SPD matrices

I've been asked to prove that the following iterations converge to the eigenvalues of SPD $A_0 \in \mathbb{R}^{n \times n}$ $A_0 = \begin{bmatrix}a & b\\ b & c \end{bmatrix}$ with $a \geq ...
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2answers
33 views

Change of basis to find coordinates

This was my previous post: http://math.stackexchange.com/posts/1243265 I changed my question quite drastically and I didn't feel I was asking the question correctly. I attempted with change of ...
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0answers
10 views

Diagonalization of an endomorphism

Let $K$ be a field of characteristic $0$ and $K[X,Y]$ be the polynomial ring in two variables. Consider the endomorphism $\Delta \in End_K (K[X,Y]$, $\Delta : f \mapsto X \cdot \delta_X (f) + Y ...
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0answers
16 views

Determining Counts of Discrete Objects Using Linear Algebra

I'm teaching myself linear algebra and was able to solve the following question using trial and error, but--how would one setup and solve a question like this using Linear Algebra? I have 32 bills ...
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1answer
23 views

Representation of a matrix as product of unitary matrices and diagonal matrix

Let $C=A+B$ where $A$ is a symmetric positive definite matrix and $B$ is a positive semi-definite skew symmetric matrix. Clearly $C$ is neither symmetric nor skew symmetric.Then is it possible to ...
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1answer
23 views

how to identify the subspace of vectors? [on hold]

Which of the following subsets of $\Bbb R^3$ are subspaces of $\Bbb R^3$? A. The $3\times 3$ matrices with all zeros in the second row B. The $3\times 3$ matrices whose entries are all integers C. ...
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0answers
24 views

How to find eigen vector for an eigen value in generalized eigen value problem

I have a generalised eigen value problem of the form $A$x = λ$B$x. I have computed the eigen value (say λ1) I am interested in using Eigen library(C++). However, because the library does not support ...
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1answer
19 views

Finding a linear map.

I have some problem with a question related to linear maps. I know the solution but I can not understand the reason behind it. For any polynomial $p∈P^2$ let: ...
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0answers
17 views

What is my error in this matrix / least squares derivation?

I'm doing a simple problem in linear algebra. It is clear that I have done something wrong, but I honestly can't see what it is. let, $y = Ax$, $y_{ls} = Ax_{ls}$ where A is skinny, and $x_{ls} = ...
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0answers
36 views

Proving the hat functions are linearly independent

$$ H_i(x) = \begin{cases} (x-x_{i-1})/(x_i-x_{i-1}), & x_{i-1}\le x\le x_i, \\ (x_{i+1}-x)/(x_{i+1}-x_i), & x_i\le x\le x_{i+1}, \\ 0, & \text{otherwise}. \end{cases} $$ How can I prove ...
5
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1answer
65 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
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1answer
23 views

Bounded operator on $L^{2}(a,b)$

Let $p\in]1,\infty[$ and consider the mapping $$ T : L^{2}(-2,2) \to L^{2}(-2,2), \quad (Tf)(x):=xf(x)$$ I want to show that $T$ is bounded, $||Tf||_L \leq T ||f||_L $. So, $$ ||Tf||_L \leq ...
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1answer
44 views

Is the Inner Product a uniformly continuous function?

I know it's continuous but is it uniformly continuous?
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0answers
31 views

How can I solve this system of linear equations?

$$\begin{align} x+y(k^2-6)+z(4k+4)&=5k+3\\ -x+y(2k^2-6)+z(4k+4)&=6k+3 \end{align}$$ I must use matrices to find for which values of $k$ this system has: exactly one solution, ...
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2answers
23 views

Point reflection uniqueness

Suppose we have normed vector space V and mapping R from V to itself satisfying the following properties: 1) R has unique fixed point $~a\in V$ 2) $\forall x\in V ~~~~ |Rx-a|=|x-a|$ 3) $\forall x ...
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1answer
50 views

Playing around with Geometric series. Where is my algebraic mistake?

Playing around with geometric series I got confused. First, consider the following equation $$\sum_{i=0}^n 1^i = n+1$$ Now, replacing $1$ by $\frac{a}{a}$ gives $$\sum_{i=0}^n ...
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0answers
19 views

How to differentiate between $(\lambda_{0}-\lambda)^{k} \,\text{and } g(\lambda) \,\text{in } f_{A}(\lambda)$?

By definition, $\lambda_{0}$ has algebraic multiplicity $k$ if $\lambda_{0}$ is a root of $f_{A}(\lambda)=(\lambda_{0}-\lambda)^{k}g(\lambda)$. What am I missing from this? ...
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64 views

Linear Algebra Proof conformation

I am trying to prove this theorem in my book. Because it is provided without proof, please let me know what you think! $\mathbf{Theorem:}$ Let V be a finite , $n$ -dimensional vector space and let U ...
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0answers
17 views

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
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36 views

When is the product of two arbitray matrices symmetric?

Let $\mathbf{A}$ be a real $n \times m$ matrix. Let $\mathbf{B}$ be a real $m \times n$ matrix. How to solve the following matrix equation? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
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3answers
54 views

Basic way to show for $n\times n$ matrices $A$ and $B$, that $(AB)^{-1} = (B^{-1})(A^{-1})$

In looking at matrix inverses, I know the following works (I is the identity matrix): If $AB$ are nxn matrices and are invertible, then $(AB)C = I$, and therefore $C = (AB)^{-1}$. I can show that ...
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1answer
33 views

Tutte matrix - Determinant

I'm trying to understand the proof of the "magic theorem" about the Tutte matrix which states: Let $T$ be the Tutte matrix of $G(V, E)$. Then, $$\det(T) = 0 \quad\Longleftrightarrow\quad G ...