Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, etc. For questions specifically concerning ...
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8 views
Maximum related to Nonnegative Matrix
Suppose $A$ is a fixed nonnegative $n\times n$ matrix (i.e. $A_{ij}\geq0$ for all $i,j$). Then for any arbitrary $n$ positive numbers $x_1,\ldots,x_n$, we denote:
...
2
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0answers
17 views
Intuiting Product of Elimination Matrices (and NOT by Matrix Multiplication)
I want to intuit, and NOT compute with matrix multiplication, $M:=\color{green}{E_{P_3 \rightarrow P_4}}\color{#CA790F}{E_{P_2 \rightarrow P_3}}E_{P_1 \rightarrow P_2},$ where:
$E_{P_1 \rightarrow ...
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votes
1answer
37 views
A basic doubt on the proof of Cauchy schwarz inequality
Consider $x$ and $y$ be any two vectors. Let $k$ be any scalar.
Now the expression $\langle x-ky,x-ky\rangle$ is a non-negative quantity and it gets minimized when $k = \frac{\langle ...
1
vote
0answers
10 views
Find a function $u(x,y)$ such that a line integral $I=u(B) - u(A)$ where B and A are limits of the integral
As the title, where function $u(x,y)$ can satisfy $I=u(B)-u(A)$
the line integral $I$ is already shown to be path independent and is defined as
...
3
votes
0answers
43 views
Derivative of trace of a matrix
$$\dfrac{\partial\operatorname{Trace}\left[\left(AB\right)^{T}Q\left(AB\right)\right]}{\partial A}=\text{ ?}$$
where $Q = Q^T>0$
2
votes
0answers
33 views
Determining the Jordan Canonical Form $18\times 18$ matrix
Let A be an $18\times 18$ matrix over $\mathbb{C}$ with characteristic polynomial equal to
$$
(x-1)^6(x-2)^6(x-3)^6
$$
and a minimal polynomial equal to
$$
(x-1)^4(x-2)^4(x-3)^3.
$$
Assume ...
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1answer
16 views
Show a class of tridiagonal matrices is nonsingular.
I'm studying for an exam and came across a problem I can't seem to solve. The problem is as follows. Show that tridiagonal matrices ...
2
votes
2answers
34 views
Prove that invertible metrices set is an open set in a given space, and the determinant is continuous [duplicate]
Given a matrix $M_{n\times m}$, we can think about it as a vector in $\mathbb{R}^{n\times m}$ (How come?).
How can I prove that the set of all the invertible metrices of size $n\times n$ is an open ...
3
votes
1answer
25 views
Automorphism in the special linear algebra $\mathfrak{sl}_2(F)$
If $L=\mathfrak{sl}(n,F), g\in GL(n,F)$, prove that the map of $L$ to itself defined by $x\rightarrow -gx^tg^{-1}$ ($x^t=$transpose of $x$) belongs to $\operatorname{Aut}L$. When $n=2,g=$identity ...
7
votes
3answers
67 views
A rational “fifth root” of the scalar matrix $2I$
I am working on the following problem from a past exam.
Find a necessary and sufficient condition for there to exist a square matrix $A$ of order $n$ whose entries are all rational, such that $A^5 ...
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0answers
17 views
When do a homogeneous linear differential equation's solutions form a vector space?
It seems "obvious" that an n th order homogeneous linear differential equation's solutions form an n-dimensional vector space, if all differentiation is with respect to one variable, say x. Of course, ...
0
votes
1answer
47 views
Intuitive meaning of right and left eigenvector
I am trying to get an intuitive understanding of the meanings of right and left eigenvectors. I guess the best thing you can do is to provide examples of application. (Examples from the field of ...
-1
votes
0answers
21 views
Linear equations containing irrational numbers [on hold]
Solve a4+a2-2squareroot3a2-square root 3a+1, if a+1/a= square root 3
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votes
2answers
31 views
Straight line test 2
A,B and C are points on a straight line that makes an angle of ceta with the positive x-axis.given that cos ceta = 4/5,B is the point (-1,5) and the coordinate of A is (t,0).Find a)the equation of the ...
0
votes
2answers
29 views
How many whole bricks 6 × 12 × 24 cm3 will be sufficient to construct a solid cube of minimum size
How many whole bricks 6 × 12 × 24 cm3 will be sufficient to construct a solid cube of minimum size
I dont know how to tackle this minimum size condition . Can someone explain this part and solve the ...
