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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Basis of the intersection of two spans

I am having troubles trying to solve this example: In $\mathbb{R}^4$, find a basis of $L1 \cap L2$, when $L_1=\operatorname{Span}\{a, b, c\}$ and $L_2=\operatorname{Span}\{d,e, f\}$ where: ...
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44 views

Spectral norm of product

If $A$ and $B$ are square matrices, then it is known that $AB$ and $BA$ have the same eigenvalues. Can the same thing be said about the spectral norms of $AB$ and $BA$? They have been the same in the ...
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2k views

Understanding the meaning of Unique Solution

General definition of System of Linear Equations says that "If The system has a unique solution, It has independent set of Equations" Consider the system of linear equations $$x-2y=-1$$ ...
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39 views

Modified rearrangement inequality..probably..

Prove that for real numbers $x_{n+1}=x_1\ge x_2\ge ...\ge x_n\ge 0$,$$\sum_{k=1}^n\frac {x_k}{x_{k+1}} \le \sum_{k=1}^n\frac {x_{k+1}}{x_k}$$Bringing LHS to RHS, we get $$\sum_{k=1}^n\frac ...
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1answer
67 views

Inequality with volume in Euclidean Space

Given an Euclidean Space and vectors $a_1, a_2, \ldots a_k, b_1, b_2, \ldots b_n$. Is it true that $V(a_1, a_2, \ldots , a_k, b_1, b_2, \ldots , b_n) \leq V(a_1, a_2, \ldots , a_k) \cdot V( b_1, b_2, ...
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208 views

Why is it easy to calculate $\operatorname{rank}(A)=n$?

when I read a paper with matrices methods, and found a difficult problem. Define matrix $A=(a_{jk})_{n\times n}$,where $$a_{jk}=\begin{cases} j+k\cdot i&j<k\\ k+j\cdot i&j>k\\ ...
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4answers
156 views

prove if $AB$ is invertible, $A,B$ is also invertible.

If $BA$ is invertible (where $A,B$ are matrix), then $A,B$ are invertible. I want to prove this theorem by not using the fact that if $BA$ is invertible, then we know that $(BA)^{-1} = ...
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0answers
95 views

Approximate diagonalization (eigendecomposition) of a non-symmetric matrix

Suppose that $A\in\mathbb R^{n\times n}$ is a non-symmetric and diagonalizable matrix with $A=PVP^{-1}$, where $P,V\in\mathbb R^{n\times n}$ and $V$ is a diagonal matrix. Here I have only an estimate ...
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3answers
66 views

Characterization of Matrices Diagonalizable by Matrices P such that P times P^Transpose is Diagonal

Let $M$ be a square matrix with complex entries. What is a characterization of $M$ such that $M = P^{T} D P$, where both $D$ and $P^{T} P$ are diagonal matrices? For example, such a characterization ...
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116 views

Given 500 parts and a list of orders, pick 50 parts to maximize the number of fulfillable orders

I'm going to start with a proclamation that this kind of optimization is new to me, so don't fault me for setting up the problem in a weird way. Please let me know if this is unclear. In a ...
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55 views

What does it mean to find a matrix of linear transformation in given basis?

I have a following problem to do: A linear transformation $f: \Bbb{R}^3 \rightarrow \Bbb{R}^2$ is defined with a formula: $$f(\mathbf{x}) =( \begin{smallmatrix} x_1+x_2\\ x_2+x_3 ...
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40 views

What is the algebraic keyword for the fact that $f$ can be split in $h\circ g$ if $\ker(g)\subset\ker(f)$

Suppose we have (possibly infinite-dimensional) vector spaces $A, B,$ and $C$. For linear maps $f\colon A\to B$ and $g\colon A\to C$ such that $\ker(g)\subset \ker(f)$, we have $f=h\circ g$ for a ...
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80 views

Conditional inequality

Let x,y,z be positive reals with $xy+yz+zx=1$. Prove the inequality $$\sum_{cyc(x,y,z)}\frac {2x(1-x^2)}{(1+x^2)^2} \le \sum_{cyc(x,y,z)} \frac x{1+x^2}.$$ I substituted $x=tan\frac{\theta}2, ...
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65 views

Book about JCF and RCF matrices

I am looking for a book in linear algebra covering both JCF and RCF matrices. Do you know a good one?
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81 views

Vector spaces and dimension: unordered pairs

Let $K= \mathbb{Z}_p$, where $p$ is a prime number, and let V be a vector space over the field K such that $\dim{V} = 3$. I have no idea where to start with this, I'm not even really sure what I'm ...
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53 views

Does the following type of “SVD” exist?

SVD of $A$ gives $U$, $\Sigma$ and $V$ such that $A = U \Sigma V$. I am interested in a different problem. Given an $A$ and $\Sigma_1,\ldots,\Sigma_{n-1}$ diagonal matrices, such that we know that ...
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1answer
81 views

Root of sum of squared distances

Say I want to calculate the euclidean distance of all edges of a triangle. I could take the root of the squared distance of each edge and add those. This would give me the right result. Adding up ...
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1answer
48 views

why do I get the following result with Matlab's SVD?

I am doing an SVD on a random matrix in Matlab $A = U S V'$. Then, I do calculate $B = U V'$. Finally, I do SVD on $B = U_2 S_2 V_2'$. I would expect $U_2 = U$ and $V_2 = V$ up to a permutation and ...
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116 views

Does every inner product space have an orthonormal basis?

I'm reading Halmos' text and he defines 'basis' as a maximal orthonormal subset of a Hilbert space $H$, but this definition seems inconsistent with the standard definition of basis. With the standard ...
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690 views

Finding vectors in a hexagon

This is a regular hexagon. I am supposed to find CD. I failed. I don't know how this happened.
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22 views

Sufficient Condition for a matrix to be null

Let $A=[a_{ij}]$ be a matrix of $M_n(\mathbb{C})$. Suppose that for any permutation $\sigma$ of the set $\{1,\ldots,n\}$ and any complex numbers $u_1,\ldots,u_n \in \mathbb{C}$ of modulus one (i.e. ...
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62 views

Is it a telescopic inequality??

Let $a_{i}>0, i=1,2,3,...,n. a_{n+1}=a_{1}$. Prove that $$2\sum_{k=1}^n \frac {a_{k}^2}{a_{k}+a_{k+1}} \ge \sum_{k=1}^n a_k$$ I took the RHS to the LHS and simplified which gave me the following ...
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144 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.9, Problem 12

If $f_1, \ldots, f_p$ are linear functionals on an $n$-dimensional vector space $X$, where $p<n$, then how to show that there is a vector $x \ne 0$ in $X$ such that $f_1(x) = 0, \ldots, f_p(x)=0$? ...
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1answer
77 views

3-variable non-symmetric inequality

Prove that for $a,b,c > 0$ $$\frac{a+b-2c}{b+c}+\frac {b+c-2a}{c+a}+\frac {c+a-2b}{a+b} >0$$ What I did is this:- Let $f(a,b,c)=\frac{a+b-2c}{b+c}+\frac {b+c-2a}{c+a}+\frac {c+a-2b}{a+b}$. ...
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2answers
334 views

Inequality between AM-GM

Prove that for $x>y>0$ $$\sqrt {xy} <\frac {x-y}{\ln x-\ln y}<\frac {x+y}2$$ Using $x=y+k$, we can turn the inequality into $$\sqrt{y^2+ky}<\frac k{\ln(1+\frac ky)}<y+\frac k2$$ Now ...
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1answer
94 views

Is the product of any two invertible diagonalizable matrices diagonalizable?

I'm studying Linear Algebra. I saw an example of a pair of 2 by 2 or n by n diagonalizable matrices, the product of which is not diagonalizable. Is there a similar example when I replace the condition ...
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111 views

Show that matrices multiplication and LUP decompositions have the same difficulty

Let $M(n)$ be the time to multiply two $n\times n$ matrices, and let $L(n)$ be the time to compute the LUP decomposition of an $n\times n$ matrix. How to show that multiplying matrices and ...
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43 views

LU factorization

Let $A$ be a nonsingular $n\times n$ matrix and suppose that Gaussian elimination with partial pivoting has been applied to produce $PA = LU$, where : - $P$ is a permutation, - $L$ is a unit lower ...
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1answer
357 views

Transpose transformation matrix with respect to the base R2x2

I found the following transformation matrix dor the transpose of a 2x2 matrix in $R^{2x2}$ (vector space of the 2x2 matrices with real numbers as elements). \begin{bmatrix} 1 & 0 & 0 ...
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27 views

What are the main uses of the LU decomposition?

I know how to perform the procedure, but why is it useful? The question is inspired by the eigenvalue decomposition, which is great for powering matrices. Thanks
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60 views

How to find the minimal polynomial of a matrix in GF(2)-Galios field

I am finding the minimal polynominal of a matrix using Berlekamp_Massey method. Let see my problem, given a matrix $M$ in galios field 2 as ...
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1answer
131 views

How to find this determinant of the binomial coefficient $\det{(A)}$

Let matrix $$A=\begin{bmatrix} \binom{m}{k}&\binom{m}{k+1}&\cdots&\binom{m}{k+n-1}\\ \binom{m+1}{k}&\binom{m+1}{k+1}&\cdots&\binom{m+1}{k+n-1}\\ ...
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1answer
177 views

Drawing a tetrahedron from a parellelepiped to convince myself it is 1/6th the volume,

I drew a parallelepiped that is spanned by three vectors, and we know the volume is given by the absolute value of the determinant of the matrix - with the three vectors arranged in rows (or columns, ...
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4answers
71 views

Choose a coefficient so that the system of equations has exactly one solution

Choose the value of $a$ from $\{-4, 88\}$ so that the system of equations has exactly one solution: $$2x +20y+3z=1$$ $$2x +2y+41z=2$$ $$ax -22y-44z=-3$$ I tried solving the system for $a=-4$ and ...
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1answer
42 views

A simple question on isometry

In Lemma 2.4 here http://link.springer.com/article/10.1007/BF01215346#page-2 why is $\sigma$ an isometry and why is $\sigma(S)^\perp=\sigma(S^\perp)$? "Lemma 2.4. Let $S$ be a linear subspace of the ...
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Multivariate polynomials with prescribed zeros

Let $P_k\subsetneq\Bbb R[x_1,\dots,x_n]$ be the set of degree $k$ multivariate real polynomials. Pick a subset $S$ of $\{-1,+1\}^n$ of size $|S|<\sum_{i=0}^k\binom{n}{i}$. We seek a polynomial ...
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1answer
47 views

Linear transformation invariant wrt. maximum.

All matrices are real. Define the operator $\max$ on matrices as a function that returns the largest value in each row. Consider a matrix $F$ of size $n \times l$. The matrix has the property that ...
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2answers
30 views

Simple question of vectors and points

Let $L_0$ be a line in $R^3$, that passes through the points $(1, 2, 3)$ $(0, 0, 0 )$ Let $L_1$ be the parallel line to $L_0$ that passes through $(3, 2, 1)$ Prove/disprove: $(1,1,1) \in L_1$ ...
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1answer
83 views

Problem understanding proof involving determinant

I have problems understanding Theorem 3.2, page 29 from Theory of Linear and Integer Programming. I don't understand (3): Let $M$ be a matrix in $\mathbb{Q}^{n\times n}$, and let $M_{ij} = ...
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21 views

inequality For the nonsingular upper triangular

Given a nonsingular upper triangular matrix $U$. Show that: $$\|U^{-1}\|_{\infty}≥\frac{1}{\min_{i}|u_{ii}|}.$$ I know $U^{-1}$ is a upper trianguler matrix that the diagonal elements of $U^{-1}$ are ...
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703 views

Relation between norm and determinant of a linear operator [closed]

Let $A$ be a $n\times n$ matrix and define $T:\mathbb R^n\to \mathbb R^n$ by $T(X)=AX$. Is there a formula that can present the norm $\|T\|$ as the determinant $\det(A)$?
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72 views

Prove that a graph which is constructed with matrices is strongly regular

Suppose that $F_q$ is a field with $q$ elements. Consider all $2\times d$ matrices with entries in $F_q$, so we have $q^{2d}$ matrices. Consider each matrix as a vertex, and two vertices $A$ and ...
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111 views

Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product: $$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$ Is the cosine of the angle between them. For an application I was ...
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220 views

Is there an example of unitary operator that is not a normal operator? [closed]

Is it possible that a unitary operator is not a normal operator?
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312 views

Shortest distance between a point and a plane using orthogonal projection

Given a point $v = (x_1,y_1,z_1) \in \mathbb R^3$, and a plane P:= $ax+by+cz=d$, find the shortest distance between $v$ and $P$. My attempt at a solution Consider $U = \{(x,y,z) \in \mathbb ...
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1answer
135 views

Trigonalise a matrix

Let $\mathbf{A}=\begin{bmatrix} 0 & 1 & 1 \\ -1 & 1 & 1 \\ -1 & 1 & 2 \end{bmatrix}$ Trigonalise a matrix Could someone trigonalise this matrix ...
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1answer
60 views

Show that $(1+a_1x+\ldots+a_rx^r)^k=1+x+x^{r+1}q(x)$

Fixed $k\ge 1$. Show that for each $r$, you can find $a_1,\cdot\cdot\cdot,a_r\in \mathbb{F}$ such that :$$(1+a_1x+\cdot\cdot\cdot+a_rx^r)^k=1+x+x^{r+1}q(x)$$ where $q(x)$ is a polynomial. Any ...
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1answer
32 views

what is an example of a normed space such that $||\sum x_i||=\sum ||x_i||$ does not imply they have the same direction?

Let $V$ be a normed space over $\mathbb{K}$ and $x_1,...,x_n\in V\setminus\{0\}$ such that $||\sum_{i=1}^n x_i||=\sum_{i=1}^n ||x_i||$. If $||\cdot||$ satisfies parallel law, then this implies that ...
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2answers
108 views

the upper triangular matrix problem

Given a nonsingular upper-triangular matrix $U$ whose diagonal elements are $u_{ii}$. Show that: the diagonal elements of $U^{−1}$ are the reciprocals of the diagonal elements of $U$. I know ...
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1answer
84 views

Matrix reduction trigonalisaton

Let $ \mathbf{A}=\begin{bmatrix} 2 & -1 & -1 \\ 2 & 1 & -2\\ 3 & -1 & -2 \end{bmatrix} $ Trigonalise a matrix in process of ...