8
votes
0answers
194 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrix $A,B,C\in M_{n}(C)$ is Hermitian matrix and is Positive definite matrices ,such $$A+B+C=I_{n}$$show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge ...
3
votes
0answers
169 views

How prove this stronger Cauchy-Schwarz inequality for traces of compression matrices

Question: Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
2
votes
2answers
29 views

Linear inequalities with one unknown

$$3x+\frac{2}{4} \leq x+\frac{7}{2}$$ What is the solution to the inequality? I started multiplying both sides by 4 which gave me $3x+2\leq14$. Then I subtracted two from both sides obtaining ...
2
votes
2answers
42 views

All unit vector has bounded components?

If $\Vert v_i\Vert \leq 1$ for all $1\leq i\leq k$ with $\{ v_1,..,v_k\}$ is linearly independent THEN FOR ALL real numbers $\alpha_i$ with $$\Vert \sum_{i=1}^k\alpha_iv_i\Vert=1$$ we can find ...
3
votes
1answer
26 views

Show $\mid\textrm{tr}(A^tB)\mid\le\sqrt{\textrm{tr}(A^tA)\cdot\textrm{tr}(B^tB)}$

Show that for all matrices $A,B\in\mathbb R^{n\times n}$ the inequality $$\mid\textrm{tr}(A^tB)\mid\le\sqrt{\textrm{tr}(A^tA)\cdot\textrm{tr}(B^tB)}$$ holds. It looks similar to Cauchy-Schwarz's ...
3
votes
1answer
27 views

Show $\textrm{Tr}(f\circ f^*)\ge 0$ for euclidian/hermitian space $(V,<,>)$

How can I show that for an euclidian/hermitian space $(V,<,>)$, for every endomorphism $f:V\to V$ and the adjoint map $f^*$ the inequality $\textrm{Tr}(f\circ f^*)\ge 0$ is valid and only when ...
0
votes
0answers
36 views

Prove solution does not exist for inequalities system

I have an inequalities sytem like the following: Example > x+y+z <= A > x+y <= B > x+z > C > y+z > D > x >= E Let A,B,C,D,E be any ...
4
votes
3answers
77 views

Matrix inequality for square matrices

Does the following hold for any square matrix $A$, $(AA^*)^{1/2}\geq (A+A^*)/2$, where superscript $*$ denotes the Hermitian transpose. Proof/any comment would be appreciated.
0
votes
0answers
22 views

Cauchy Schwarz inequality with an operator

The standard Cauchy-Schwarz inequality is given by, $|\langle\Phi|\Psi\rangle|^2\le\langle\Phi|\Phi\rangle\langle\Psi|\Psi\rangle$ But now I'm intressted in what happens to ...
12
votes
3answers
185 views

determinant inequality $\det(A^2+AB+B^2)\geq\det(AB-BA)$

$A,B$ are two $2\times 2$ real matrices, then $$\det(A^2+AB+B^2)\geq\det(AB-BA)$$ The inequality is equivalent to the following problem: Let $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
0
votes
1answer
15 views

Inequality on lengths and sums of vectors $\left\lVert\sum_i \vec{a_i}\right\rVert \le \sum_i \left\lVert \vec{a_i}\right\rVert$

I'm trying to show the following inequality, which expresses the fact that the magnitude of the sum of some vectors, is less than the sum of the individual magnitudes: $$\left\lVert\sum_i ...
0
votes
0answers
18 views

Simplifying inequalities given some assumptions

Say I have an expression of conjunct and disjunct inequalities. $$ 10 x+5 y-100>0\;\land\\ 10 x+5 y-200<0\;\land\\ -10 x+5 y-100>0\;\land\\ -10 x+5 y-200<0\;\land\\ ...
0
votes
1answer
36 views

Matrix Inequality: $A^\top B A \preccurlyeq B$

Consider $A, B \in \mathbb{R}^{n \times n}$, $A$ invertible, $B \succ 0$. Say if the following holds: $$ A^{- \top} B A^{-1} \preccurlyeq B \ \Longleftrightarrow \ B \preccurlyeq A^\top B A. $$ I do ...
1
vote
2answers
116 views

Eigenvalue inequality $\lambda_{\min}(AB) \geq \lambda_{\min}(A)\lambda_{\min}(B)$

$A, B$ are $n\times n$ positive semidefinite Hermite matrices. $\lambda_{\min}(A)$ denotes the minimum eigenvalue of $A$. then we have $$\lambda_{\min}(AB) \geq ...
1
vote
1answer
44 views

Inequality of Weighted norm

I have a question about the weighted norm inequality: The weighted norm of a vector $x\in R^{M\times N}$ is defined by: $\left \| X\right \|_{w,*} = \sum_{_{i}}\left |w_{i}\sigma _{i}\left ( X ...
2
votes
1answer
54 views

Elementary matrix inequality

Let $A \in \mathbb{R}^{n \times n}$ be a positive semidefinite matrix. Is the mapping $$ \begin{align} F \ \colon \ \mathbb{R}^{n \times n} &\to \mathbb{R}^{n \times n} \\ X &\mapsto X^{-1} - ...
2
votes
2answers
31 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
4
votes
1answer
127 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
1
vote
1answer
84 views

Cauchy-Schwarz Inequality - Proof using Quadratic Polynomial [Lay P379 Thm 6.7.16]

I don't perceive https://www.dpmms.cam.ac.uk/~wtg10/csineq.html, about why it " is an obvious thing to write down" "a quadratic form, use the fact that it is non-negative everywhere, and ...
2
votes
1answer
81 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
3
votes
1answer
102 views

counterexamples to $ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) $

$n\geq3$. A and B are two $n\times n$ reals matrices. For $n\times n$, Could one give counterexamples to show that $$ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) \tag{$*$}$$ is not necessarily true? ...
9
votes
2answers
248 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
0
votes
1answer
51 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
1
vote
1answer
33 views

Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...
1
vote
2answers
42 views

Number of solutions for inqeuality

Is there a way we can determine number of solutions for equation $$x*y < d$$ where d is constant and x & y are positive integers greater than 1. I am not interested in actual values, but ...
2
votes
3answers
39 views

Find such anti-symmetric matrix $W$ that $A^T WP \geq 0$

$P$ and $A$ are both n-dimensional vectors with non-negative components. $W$ is an $n\times n$ matrix with $W_{ij}=w_i-w_j$, where all $w_k\geq 0$. So $W$ is an anti-symmetric matrix with some ...
1
vote
1answer
65 views

Number of solutions for 2 equations involving 4 variables

Given that $a, b, c, d$ are positive integers, What are the number of solutions for the given 2 equations, $\mathbf{ad - bc > 0}$ $\mathbf{a + d = n }$ where, $n$ is a given positive integer.
1
vote
1answer
28 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
1
vote
1answer
38 views

Solving inequalities with absolute values on both sides

I need to find the solution sets for the following inequalities: $$|3+2x|\leq|4-x|$$$$|2x-1|+|1-x|\geq3$$ After a bit of tinkering with the first one, I think the solution set is $[-7, \frac13]$, ...
1
vote
3answers
56 views

Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
0
votes
0answers
25 views

Compound inequalities with separate variables

So I've got to find the set of elements for which the following inequality holds true: $$x-2<12\leq6-5x$$ I've only been taught to solve these kinds of inequalities with one x in the middle, so I ...
7
votes
1answer
237 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
0
votes
1answer
38 views

Proving triangle inequality geometrically

If $\mathbf{u}$ and $\mathbf{v}$ are vectors, how can you prove geometrically that $$ \|\mathbf{u}+\mathbf{v}\|\leq\|\mathbf{u}\|+\|\mathbf{v}\| $$ I am aware of the proof that uses dot product but I ...
2
votes
2answers
48 views

2x2 Matrix Inequality

Is it true that if I have a positive definite matrix $m = \left( \begin{smallmatrix} m_{11} & m_{12}\\ m_{21} & m_{22} \\\end{smallmatrix} \right)$ in $\mathrm{M}(\mathbb{C};2)$ the following ...
1
vote
2answers
92 views

Solving inequalities with fractions with unfactorable polynomials

So I've been cracking my head open trying to solve this inequality: $$\frac {x+1}{2-x} \le \frac {x}{3+x}$$ I've been taught you have to put all factors to one side of the inequality (leaving zero ...
3
votes
1answer
127 views

A determinant inequality

Let $A,B$ be two $m\times n$ real matrices. Then $$|AA'|\cdot |BB'|\geq |AB'|^2.$$ For square matrices, it is the equality. How to prove this inequality then?
0
votes
0answers
68 views

Solve the system of inequalities. Optimization problem.

I have a set of linear inequalities as follow: ...
3
votes
1answer
38 views

Prove unequal with numbers in $\mathbb{R}^{+}$

I have problems to prove the following inequality: Let $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$ $\in$ $\mathbb{R}^{+}$ then: $$\left( \frac{b_{1}+b_{2}}{a_{1}+a_{2}}\right)^{a_{1}+a_{2}}>\left( ...
1
vote
0answers
50 views

Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
0
votes
2answers
52 views

Norm of a matrix and lower bound for its determinant

Assume that $M$ is a positive constant, $A=[a_{ij}]$ is a matrix, and $\vert a_{ij}\vert \geq M $ for all $1\leq i,j \leq n$. Also, assume that $\det(A) \neq 0$ .Can we conclude that there exists a ...
1
vote
1answer
40 views

Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
3
votes
0answers
44 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
1
vote
3answers
49 views

A question on inequalities

What is the solution set of the inequality $$ \frac{2x - 1 }{x+1}\lt0$$ One answer that is quite simple to get is $$x\lt1/2 $$ What can be the other value for the solution set...??
1
vote
2answers
134 views

Proving AM ≥ GM in 3 variables using the Cauchy-Schwarz inequality

This question is from the textbook Introduction to Linear Algebra by Gilbert Strang, where the author has asked to prove that $$\sqrt[3]{xyz} \le \frac{x+y+z}{3}. $$ The only equations at hand are ...
1
vote
1answer
64 views

Relation between softmax and max

For two vectors $X$ and $Y$ in $\mathbf{R}^n$, does the inequality below hold? $\left| \text{softmax} X - \text{softmax} Y \right| \leq \text{max} | X - Y |$ Softmax is the same as log-sum-exp: ...
0
votes
1answer
470 views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
2
votes
2answers
106 views

Positive matrix and positive vector

Let $A \in \mathbb{R}^{n \times n}$ be a non-negative matrix, i.e. $A_{i,j} \geq 0$ $\forall (i,j)$. Let $x \in \mathbb{R}^n \setminus \{0\}$ be a non-negative vector, i.e. $x_i \geq 0$ $\forall i$. ...
1
vote
2answers
61 views

Inequality regarding norm vector space

I am not sure how to prove this inequality involving norms. Let $X$ be a normed vector space and $x,y$ are vectors in $X$ with nonzero norms. Prove the following inequality is true. $$\|x-y\|\geq ...
0
votes
0answers
48 views

Given a set of inequalities, use what algorithm to minimize one variable?

I have the following set of inequalities: $$C_1 \le y-T_1x_1 \lt C_1+D_1$$ $$C_2 \le y-T_2x_2 \lt C_2+D_2$$ $$C_3 \le y-T_3x_3 \lt C_3+D_3$$ where $x_1, x_2, x_3$ and $y$ are non-negative integer ...
0
votes
2answers
35 views

Question on inequalities. How to show if $a+b+c+d+e\le r+s+t+u+v$ and $a-r\ge0, b-s\ge0,c-t\ge0,d-u\ge0,e-v\ge0$ implies $a=r, b=s,c=t,d=u,e=v$

Question on inequalities. How to show if : $a+b+c+d+e\leq r+s+t+u+v$ and $a-r \geq 0, b-s \geq 0,c-t \geq 0,d-u \geq 0,e-v \geq 0$, then $a=r, b=s,c=t,d=u,e=v$. How do you show this is true. I can ...