Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Affinely independent

I am wondering about affinely independent and just linearly independent. On Wikipedia it is explained that $u_i$ are affinely independent if $u_1 - u_0, ...,u_k -u_0$ are linearly independent. It is ...
2
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1answer
48 views

Gradient of $\displaystyle \left|\sum_{k=1}^Nx_k^2e^{-j\frac{2\pi}Nkl}\right|$?

How to evaluate gradient of this function ? $$\displaystyle f(\mathbf{x}) = \sum_{l=1}^{N-1} \left(|\sum_{k=1}^Nx_k^2e^{-j\frac{2\pi}Nkl}| - A\sum_{k=1}^Nx_k^2\right)^2 $$ $\mathbf{x}$ is a real ...
7
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1answer
465 views

Determinant game - winning strategy

I came across this problem while looking at Putnam problems a while ago: "Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008 x 2008 array. Alan plays ...
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1answer
60 views

Write the normal and vector form of the equation in $\mathbb{R}^2$

This is more of a check then anything else. Here is what I have. Need to find the normal and vector form of the equation $$-2x+3y=5$$ Normal form: $$(-2,3) \cdot [(x,y) - (-1,1)]$$ Vector form: Now ...
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107 views

What is wrong with this thinking? Linear Algebra problem

Suppose $T \in L(V)$ such that each vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity. I have googled the answer with my own proof and mine looked almost ...
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2answers
137 views

Freshman's dream in the quotient

Let $k$ be a field of characteristic $p$ and let $A$ be a $k$-algebra. Let $S$ be the subspace of $A$ generated by the commutators, that is, the $k$-span of elements of the form $[a,b] = ab-ba$ (I ...
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1answer
622 views

Orthogonal projection onto an affine subspace

If we want to find the distance from a vector $x$ to a subspace $S$, we take $\| (I-P_S) x\|$, where $P_S$ is the orthogonal projection onto the subspace $S$. Obviously we could do the same thing for ...
2
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1answer
142 views

Prove a matrix is positive definite

Please, can somebody help me with this problem? [Ciarlet 5.3-1] Let $A$ be an invertible Hermitian matrix, with the splitting $A = M-N$, $M$ being an invertible matrix. Prove that, if the Hermitian ...
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2answers
152 views

ISO brief primer on special matrices

I am looking for a brief primer on the following types of matrices: stochastic, doubly stochastic, symplectic, Vandermonde, Hadamard, permutation, tridiagonal and circulant. Nothing too deep, just a ...
2
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4answers
866 views

Do all symmetric $n\times n$ invertible matrices have a square root matrix?

My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with ...
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1answer
117 views

Finding a point within a 2D triangle

I'm not sure how to approach the following problem and would love some help, thanks! I have a two-dimensional triangle ABC for which I know the cartesian coordinates of points $A$, $B$ and $C$. I am ...
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1answer
121 views

Is this isomorphism natural?

Suppose I constructed a linear map $\phi$ without choosing a basis, but in order to check that $\phi$ is an isomorphism, I chose a basis. Is $\phi$ still considered a natural isomorphism? Edit: The ...
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197 views

Commuting self adjoint operators

I want to prove that two commuting, self adjoint operators $A,B$ on a finite dimensional complex inner product space $V$ have identical eigenspaces. So far I have that the eigenspaces of $A$ are ...
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1answer
93 views

$\forall v\in V:(Tv,v)=0\implies T^{\star}=-T$

Let $V$ be a real inner product space and $T:V\to V$ a linear transformation. $$\forall v\in V, (Tv,v)=0\implies T^{\star}=-T$$ "An attempt": $$(Tv,v)=(v,T^{\ast}v)=(T^{\ast}v,v)$$
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42 views

Solving multiple linear equations

I'm a bit rusty on my linear algebra. I have the following equations: $$\operatorname{weight}_C = \frac{\frac{P_y - A_y}{B_y - A_y} - \frac{P_x - A_x}{B_x - A_x}}{ \frac{A_x-C_x}{B_x-A_x} - ...
0
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48 views

About $2\times 2$ similar matrices…

Let $A$ and $B$ be $2\times 2$ matrices with the same trace and the same determinant. Are $A$ similar to $B$? I know that they have the same characteristic polynomial. So, exist $P,Q\in GL_2(F)$ ...
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2answers
137 views

Given characteristic polynomial of $T$, need find characteristic polynomial of $T^3$

Let $T:\mathbb{R}^2\to \mathbb{R}^2 $ be a linear transformation with characteristic polynomial $x^2+2x-3$. Find the characteristic polynomial of $T^3$. How to do this? Thanks!
2
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1answer
240 views

Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: \begin{equation} ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| ...
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3answers
136 views

True or false: If $||Tv+v ||=||Tv||+||v||$, then $1$ is eigenvalue of $T$

Let $V$ be an inner product space over $\mathbb{C}$. And let $T:V\to V$ be an unitary transformation. Suppose that for $ 0 \neq v\in V$ we have $||Tv+v ||=||Tv||+||v||$, then $1$ is eigenvalue of ...
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1answer
40 views

Linear transformation with $T^3+I=0$

Is it true or false that if $T:V\to V$ is linear transformation such that $T^3+I=0$, then $\dim V\geq 3$?
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149 views

Linear Algebra 2: True or False question.

Let $A\in M_3(\mathbb{C})$. Suppose that $A^{\star}A=AA^{\star}$. Is the following true or not: If $(1,0,1)^{T}$ and $(1,1,0)^{T}$ are eigenvectors of $A$ with eigenvalues $\alpha,\beta$, then does ...
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1answer
42 views

Bilinear Form - Proof

I have to prove that the mapping $f(x,y) = {\displaystyle \sum_{i=1} ^ {n} }{ \displaystyle \sum_{j=1}^{n} }x_iy_j{f}(e_i,e_j)$ is a bilinear form, that is, inter alia, the condition: ...
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0answers
184 views

Matrix spectral decomposition

Let $A$ be a square matrix $(N \times N)$ and $a_{ij} \in \mathbb{R}$. Suppose A has N eigenvalues $\lambda_{1} < \lambda_{2} < ...\lambda_{n} \in \mathbb{R}$. $A$ = $R \Omega R^{-1}$ its ...
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2answers
72 views

bilinear form - proof

I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$ is a bilinear form, that is, inter alia, the condition: ...
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2answers
840 views

How to calculate weight of positive and negative values.

We have used formula formula to calculate weight as, $$ w_1 = \frac{s_1}{s_1 + s_2 + s_3};$$ $$ w_2 = \frac{s_2}{s_1 + s_2 + s_3};$$ $$ w_3 = \frac{s_3}{s_1 + s_2 + s_3};$$ However, their is ...
2
votes
2answers
127 views

Preimage of Lebesgue null set under a singular linear map

Let $T:\mathbb{R}^n\to \mathbb{R}^n$ ($n\in\mathbb{Z}_+$) be a linear map, but not necessarily invertible. Let $N\subset\mathbb{R}^n$ be a Lebesgue measurable null set. Now, is it true that the ...
3
votes
1answer
51 views

Is there a matrix decomposition $P = AA^{+}$, given P?

Suppose one could experimentally obtain $P$, a $N\times N$ matrix. Is there a way to decompose this into two matrices $AA^{+}$, where $A$ is $N\times M$ and $A^{+}$ is the pseudo-inverse of $A$? ...
3
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1answer
42 views

For nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, is the length of the projection of $v$ along $u$ always less than the length of $v$.

For all nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, the length of the projection of $v$ along $u$ is less than the length of $v$. This is a true/false question and when I said true it ...
3
votes
1answer
69 views

Showing that the zero vector has norm zero

I need to show that this is a property of a norm. I know this is supposed to be straightforward but I am somehow not seeing it. The property is $$\lVert 0\rVert = 0$$ I was trying to use the fact ...
0
votes
1answer
82 views

0 eigenvalue of weighted laplacian

I consider (weighted) directed graph and eigenvalues of its laplacian matrix. If a graph contains rooted out-branching which is the subgraph possessing a node can approaching to any nodes in the ...
3
votes
3answers
859 views

Is matrix diagonalization unique?

From the following statement, it seems matrix diagonalization is just eigen decomposition. Diagonalizing a matrix is also equivalent to finding the matrix's eigenvalues, which turn out to be ...
3
votes
1answer
92 views

Identity makes every matrix invertible?

I have found this in a proof and do not understand where this comes from: If A is singular, then there exists $\delta \in \mathbb{R}_{>0} \forall \epsilon\in (0,\delta): \epsilon ...
2
votes
4answers
130 views

Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions.

How can i solve this, Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions. I thought let $\frac ...
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0answers
62 views

What sequence has this Discrete Fourier Transform?

Suppose $$ x[n]= \begin{cases} x_i &, i \in P\\ 0 &, i \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_i \geq 0$. Suppose these equalities hold : $$ ...
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1answer
358 views

Is W a subspace of $\mathbb{P}_3$?

$W=\{p(x)\in \mathbb{P}_3 \mid p(-1)=p(2)=0\}.\\ $Is $W$ a subspace of $\mathbb{P}_3$? Note that $\mathbb{P}_3$ denotes the vector space of all polynomials with degree of 3 or lower. I'm not ...
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2answers
90 views

n is+ve integer, how many solutions $(x,y)$ exist for $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ with $x$, $y$ being positive integers and $(x \neq y)$ [duplicate]

I wanted to know, how can i solve this. For a given positive integer n, how many solutions $(x,y)$ exist for $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ with $x$ and $y$ being positive integers and $(x ...
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2answers
43 views

Proof that if $n<k$ and $A$ is an $n\times k$ matrix, then $A^{T}A$ is not invertible

Can I get a proof of the fact that if $n<k$ and $A$ is an $n\times k$ matrix, then $A^{T}A$ is not invertible?
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1answer
357 views

Cholesky/LU decomposition from matrix and its inverse?

Usually, we have a matrix $A$ and want to calculate the $LU$ (or sometimes Cholesky, depending on $A$'s properties) decomposition. This is often the hard part. Now, if we have the $LU$ decomposition ...
2
votes
1answer
96 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq ...
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2answers
46 views

Do we have $x^TDAx\ge \min(\lambda_D)\min(\lambda_A)x^Tx$ if $A$ is PD and D is both diagonal and PD?

Suppose matrix $A\in\mathbb{R}^{n\times n}$ is symmetric positive definite and $D\in\mathbb{R}^{n\times n}$ is both diagonal and positive definite, do we have the following result? $$x^TDAx\ge ...
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2answers
86 views

This set of matrices is dense

I'm trying to prove the set of the matrices whose eigenvalues have non-zero real part is an dense subset of $M^n$, the set of square matrices with order $n$ which is identify with $\mathbb R^{n^2}$. ...
2
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2answers
344 views

Need help calculating this determinant using induction

This is the determinant of a matrix of ($n \times n$) that needs to be calculated: \begin{pmatrix} 3 &2 &0 &0 &\cdots &0 &0 &0 &0\\ 1 &3 &2 &0 &\cdots ...
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0answers
57 views

Is there any solution for this over-determined system of equations?

Under what condition(s) a unique solution is available for this over-determined system of equations? $$ x^TA_1x = x^TA_2x = \cdots = x^TA_{N-1}x $$ where $$ A_d = [w^{d(p_i-p_j)}]_{K \times K} ...
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1answer
71 views

Problem with equation derivation

I'm studying support vector machines and the book I'm using states in one particular part the following: We know that: $$\vec{y} = \vec{x} + v\vec{w}$$ and that $$|\;\vec{y}-\vec{x}\;| = 2M$$ and so ...
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3answers
333 views

Similar matrices properties

So I have a question which I can not solve. Assuming $A,B \in \mathbb{M_{n}(\mathbb{R})}$, $A$ similar to $B$, is it possible that $\det(A) = \det(B^{2})+1$? We know that there exists $P$ ...
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1answer
66 views

Is there a good strategy for computing eigenspace corresponding to $1$ of a matrix with entries of trigonom

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
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2answers
81 views

Is there any good strategy for computing null space of a matrix with entries $\cos x$ and $\sin x$?

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
3
votes
2answers
108 views

Can I say “since $\operatorname{Char}R\ne2$ then $b=0$, hence $a=b=c=0$”?

I am trying to show that since the set $\{x_1, x_2, x_3\}$ in a vector space $V$ over a division ring $R$ is linearly independent, then the set $\{x_1+x_2, x_2+x_3, x_1+x_3\}$ is also linearly ...
3
votes
1answer
354 views

Vector Spaces and Polynomial Rings

در تعریف فضای برداری در ویژگی آخر داریم به ازای هر ایکس متعلق به فضای برداری و1 متعلق به میدان داریم یک متعلق به میدان ضرب در ایکس متعلق به فضای برداری برابر باایکس دنبال ی مثال از ش می گردم تو تابع ...
4
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4answers
116 views

Simple looking log problem

How would I solve this for $x$? The original problem is $$x+x^{\log_{2}3}=x^{\log_{2}5}$$ I have tried to reduce it down to this, $$x^{\log_{10}3}+x^{\log_{10}2}=x^{\log_{10}5}$$ I have been ...