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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2
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Is it possible to have $A \succeq B \succeq 0$ with $A^2 - B^2$ having $n-1$ negative eigenvalues?

For any dimension $n$, can we write down two symmetric, positive semi-definite matrices $A,B$ with $A \succeq B$ in the sense of the usual ordering (i.e., $A-B$ is positive semidefinite) such that $A^...
1
vote
0answers
19 views

minimize the matrix 2-norm

I'm reading a paper, and encounter some problem: Given $\left\{x_i\in\mathbb{R}^n\mid i=1,2,\ldots,N \right\}$, and $V\in M(n,p)$ is a matrix. Then what is the $A\in M(p,n)$ which would minimize $\...
0
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1answer
31 views

Using Cayley-Hamilton theorem to get a formula for $A^{-1}$ from $\chi_A$

I'm trying to prove that if an invertible n-by-n matrix $A$ has characteristic polynomial $$\chi_A(t)=(-1)^nt^n+a_{n-1}t^{n-1}+\ldots+a_2t^2+a_1t+a_0$$ with $a_0\not=0$then $$A^{-1}=\frac{-1}{a_0}((-1)...
2
votes
0answers
17 views

Complexity of computing a product of matrices

If $A$ is a non-singular $n\times n$ matrix, $B$ is an $n\times p$ matrix, and $C$ is a $p\times n$ matrix (where $1\le p \ll n$), how does one prove that the complexity of $$D=A^{-1}(BC)$$ is $\frac{...
2
votes
2answers
18 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $$0 \le a_{j,j} \le 1$$ and $$-...
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Prove that $\sqrt { 2 } +\sqrt { 3 } +\sqrt { 5 } +\sqrt { 7 } +\sqrt { 11 } +\sqrt { 13 } +\sqrt { 17 } $is irrational number? [duplicate]

I got this solution, but I didn't understand it. Assume that $b_1,b_2,b_3,\ldots, b_n$ are whole numbers (not zero). So, we have $b_1\sqrt{a_1}+b_2 \sqrt{a_2}+\ldots+b_n\sqrt{a_n}=0$. Prove it with ...
0
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1answer
14 views

Matrix equivalence via orthogonal matrices

Let $A,B \in M_n(\mathbb{R})$. We say they are equivalent if there are $P,Q$ invertible such that $A=QBP$ (note this is weaker than similarity). Every matrix is equivalent to a diagonal matrix using ...
0
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Gram matrix for a random variable vector space with inner product?

I am wondering if it is possible to construct a list of binary valued random variables, $\{\bf{X}_1,\bf{X}_2,\bf{X}_3\}$ and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\...
3
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31 views

What can be said about $x^TAx$ is terms of $\|x\|$?

Let $A \in \mathbb{R}^{n \times n}$ and $x \in \mathbb{R}^n$ What can be said about $x^TAx$ is terms of $\|x\|$? I know that from C-S inequailty, $|x^TAx| \leq \|x\| \|Ax\|$, can I go further?
0
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1answer
30 views

I'm having trouble understanding what this problem is asking me

This is the problem So my problem is that I dont know how to solve it... I have learned about system of inequalities and that kinda stuff, but I never got anything like this. I do not want anyone to ...
1
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1answer
24 views

Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
3
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0answers
15 views

Whether a given algebra is the algebra of endomorphisms for a vector space.

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
0
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1answer
16 views

How do I compute the area of this parallelogram

Given vectors $a,b$ and the ribs of parallelogram are $2a +3b = A$, $a-2b = B$. Also given $a \times b = (-1,2,2)$. Compute the surface of the parallelogram. I'm not sure where I saw but I think it ...
0
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0answers
16 views

commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
0
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1answer
16 views

Showing properties of a function and its inverse image

I tried proving the following question but did not get too far. Let $\ f:A \to B$ be a function and $\ f^{-1}(Y)$ be the inverse image of $\ Y\subseteq B$ on $\ f$. Consider the following ...
0
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0answers
27 views

Functional equation of $f(n)=\sum_{k=0}^{n-1}g\left(x+\frac{k\pi}{n}\right)$

Suppose the function $f(n)$ is given by: $$f(n)=\sum_{k=0}^{n-1}g\left(x+\frac{k\pi}{n}\right)$$ Where $x\in\mathbb{R}$. I am looking for a formula that enables me to express $f(n)$ as : $$f(n)=\sum ...
0
votes
0answers
17 views

SVD with degenerate singular values

I'm using SVD to do some kind of low rank approximation, basically I have to compute the largest eigenvalues, I also tried different routines, ...
0
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0answers
14 views

Preposition about the Entries of the Product of Markov Matrices.

Definition: A Markov matrix is an $n \times n$ complex matrix with the sum of the elements in every column equal to 1. My task is to prove that: If A, B are Markov matrices such that $|a_{ij}|\leq1$ ...
2
votes
1answer
56 views

Is a vector space isomorphic to the kernel $\oplus$ image of a map out of it?

Let $f:V\to W$ be a linear map of finite-dimensional vector spaces. By simply counting dimensions and using rank-nullity, it is clear that $V\cong \mathrm{im}\,f\oplus\mathrm{ker}\,f$. I want to know ...
0
votes
1answer
45 views

Find $B(B^{T}B)^{-1}B^{T}$.

To find: $$B(B^{T}B)^{-1}B^{T}$$ for $B=[0,1,-1]^T$ I have $$\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \left ([0,1,-1]\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \right )^{-1}[0,1,-1]$$ but ...
0
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2answers
54 views

Is the finite dimension of a vector space over the complex numbers half the dimension of the same vector space considered over the reals?

Consider a vector space V with basis ${b_{1},..,b_{n}}$ and complex scalars. This obviously has dimension n. Now consider a space with the same exact set of vectors of V, except with real scalars. I ...
0
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1answer
18 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
0
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1answer
25 views

Least squares solutions and orthogonal projection?

I found the least squares solution for the following inconsistent system of equations: $ x_1 - x_2 = 0$ $ x_1 + x_2 = 5 $ $-x_1 + x_2 = 2$ , which turned out to be $ \begin{bmatrix} 2\\ 3\\ \end{...
0
votes
1answer
15 views

Write summation of vector outer products into matrix form

My question is as follows: Given the weighted summation of vector outer products $\sum_i\sum_jh_{ij}{\bf v_i}{\bf u_j}^T$, where $h_{ij}$ is the weight, and ${\bf v_i,u_j}$ are column vectors, I was ...
2
votes
1answer
56 views

Finding non-trivial solutions for the system of linear algebraic equations

Suppose we have a system of $n$ linear algebraic equations where $n>1$ is a positive odd integer. The matrix $A=\{a_{ij}\}_{i,j=1}^n$ of this system has the following properties: $a_{ii}=0$ for ...
0
votes
1answer
13 views

Conjugacy of Cartan subalgebras

This is probabably a very silly question, stemming from some fundamental misunderstanding I have of the relevant definitions, but I am stumped by it. I know that any two Cartan subalgebras of $\...
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0answers
40 views

Don't know how to enter this into webwork [on hold]

I know the vectors are (-3-i ; 2) and (-3+i ; 2) however no matter which way I enter it into the program, it regards my answer as incorrect. How am I to enter the answer?
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0answers
28 views

factors sum to 1

If I have factors of linear operators say $$(a_1 + A)(a_2 + A)(a_3 + A)\cdots(a_n + A) = 0$$ $A$ being an linear operator(i guess it really doesn't matter its operator or not) why $$\sum_{n} \frac{...
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votes
2answers
17 views

Finding an approximate function using orthonormal basis

I'm trying to take a function in $C_0[0,1]$ space (let's call this $f(x)$) and trying to find the best approximate of $f(x)$ at $P_2[0,1]$ space (let's call this approximate $p(x)$). Note that $P_2[0,...
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0answers
10 views

Completing a semi-vector space to a vector space

If we have a semi-vector space $U$ (as defined here), what do we have to additionally demand from $U$ such that we can complete it to a vector space $\tilde{U}$ via $U\xrightarrow{\iota}\tilde{U}$ and ...
2
votes
1answer
52 views

How do I prove this statement in linear algebra?

I had a test about a week ago and I want to know the answer to this question: Given an orthonormal basis $B$ spanned by $\{ v_1, v_2 , v_3 \}$ of ${\mathbb R}^3$ Prove that for every $v \in {\mathbb ...
3
votes
3answers
33 views

Smallest possible value of the norm?

The vectors $ \vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix} $ and $ \vec{u_2} = \begin{bmatrix} 1 \\ -1 \\ 1\\ -1 \end{bmatrix} $ are orthonormal in $ \mathbb{R}^4$....
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0answers
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Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
0
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1answer
36 views

Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
2
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0answers
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Generalise Expression to slice Circulant Matrix

Suppose I have $4 \times 4 $ circulant matrix , $$A=A(0:3,0:3)=A(:,:)=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \...
1
vote
2answers
45 views

Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
2
votes
1answer
57 views

Why is this matrix symmetric?

There is an example in the Convex Optimization lecture notes, Boyd. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no ...
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0answers
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Entanglement of 3-qubit states

Given a separable 3-qubit state φ = φ0 ⊗ φ1 ⊗ φ2 with φi= ai0|0> + ai1|1>, |0>, |1> being the computational base. φ thus can be written as φ = b000|000> + b001|001> + b010|010> + b011|011> + ...
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Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
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0answers
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Calculating the coefficients of a separable 2-qubit state

Given a separable 2-qubit state φ = φ0 ⊗ φ1 with φi= ai0|0> + ai1|1> φ thus can be written as φ = b00|00> + b01|01> + b10|10> + b11|11> with bij = a0ia1j. Now let some bij be given, i.e....
2
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0answers
35 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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votes
3answers
132 views

What is the number of subspaces of a particular dimension?

If we have vector space $V$ with dimension $n$ then how many subspaces of $V$ with dimension $m<n$ are there? In my opinion the answer should be the number of ways to choose $m$ linearly ...
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0answers
20 views

sending basissen

Lets say we have this $3\times3$ matrix: $$ \begin{bmatrix} 4 &−4 &12\\ 1& -1& 3\\ −1& 1 &−1 \end{bmatrix} $$ What is the algorithm to find a basis of $\Bbb R^3$ for which ...
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Linear transformation and projection [on hold]

1 Suppose that W is a subspace of a finite-dimensional vector space V. (a) Prove that there exists a subspace W' and a function T:V→V such that T is a projection W along W'. (b) Give an example of a ...
2
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1answer
33 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
0
votes
1answer
19 views

Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
2
votes
3answers
60 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
1
vote
2answers
45 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
0
votes
0answers
25 views

problem solving in arithmetic

I've been given the following problem: The formula to find $Y$ is $Y=x_1+x_2+x_3+x_4-x_5$ The value of $Y$ is given as $100$. Now the question is: Is it possible to find without ambiguity $x_1$ ...
0
votes
1answer
47 views

$A^2$ is bounded $\implies$ $A$ is bounded?

Let $A_n$ be a sequence of $k \times k$ real matrices. Assume $A_n^2$ is bounded w.r.t some norm. Is $A_n$ also bounded? I was able to show this is true if $A_n$ are symmetric matrices (using SVD). ...