# Tagged Questions

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### 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 ...
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### 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 ...
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### Square matrix and determinant inequality

Let $A, B, C$ be invertible $n \times n$ square matrices with $AC=CA$ and $B^2C^2=I_n$ Is $\det(ABC +CBA +A^2+I_n)$ always $\geq 0$?
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### find conditions on input data such that a linear system has (no) feasible points

As a result of the apllication of Farkas' lemma I obtained the following problem: Let $m,n,q \in \mathbb{N}$, $b \in \mathbb{N}^m, l \in \mathbb{N}^m$ with $l_i \mid q$ for all $i=1,\ldots,m$. ...
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### How prove this matrix inequality $\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$

Question: Let $A_{n\times n},B_{n\times n}$ is positive Hermite matrix show that $$\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$$ I know this $$\det(A+B)\ge \det(A)+\det(B)$$ But My problem I ...
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### Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
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### Bound for Arithmetic Harmonic mean inequality for matrices?

This comes from my research in econometrics, but it boils down to a pure matrix-algebra question. The framework is as follows: We have a cross-sectional i.i.d. sample $\{\mathbf y, \mathbf X\}$, where ...
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### Why does the Cauchy-Schwarz inequality hold in any inner product space?

I am working through linear algebra problems in Apostol's Calculus, and he has numerous problems that seem to imply that Cauchy-Schwarz holds no matter how the inner product is defined. Then, he has ...
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### Apostol question on alternative definition of dot product

The problem says: Suppose we define the dot product by $A\cdot B = \sum_{k=1}^n |a_kb_k|$. Which of the following properties hold with this new defition? Does the Cauchy-Schwarz inequality still ...
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### Inequalities for Differences of Absolute Values of matrices

Let $A$ and $B$ be two real symmetric $n\times n$ matrices. Let $A=USU^T$ be the eigen-decomposition $A$ and let $|A|=U|S|U^T$ where $|S|$ just denotes elementwise absolute value of the diagonal ...
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### Certain matrix inequalities

I want to solve the following inequalities: \left| Tr\left( \frac{(X\otimes Y).A.(X\otimes Y)^*.B}{Tr((X\otimes Y).A.(X\otimes Y)^*)}\right)\right|>2, \quad\text{given} \quad ...
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### Proof of Nonnegativity Inequality

Prove the Inequality: $$\sum_{i,j}\left ( (PAQ)_{i,j}\frac{B_{i,j}^2}{A_{i,j}}- (PBQ)_{i,j}B_{i,j}\right ) \geqslant 0$$ Given that: $P$ and $Q$ are $n$x$n$ and $m$x$m$ symmetric matrices, $A$ ...
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### determine x in $x\log_\frac{1}{10}(x^2+x+1)>0$

I wanted to know, how can i determine the values of x for which $x\log_\frac{1}{10}(x^2+x+1)>0$ going to the question, we must have $x>0$ and $\log_\frac{1}{10}(x^2+x+1)>0$ or both must ...
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I am trying to prove the following result. Let $d$ be a vector in $\mathbf{R}^{n}$ with $\|d\|_{\infty} < 1$. Then, $$\sum_{i=1}^{n} \log(1 + d_{i}) \geq \mathbf{1}^{T} d - \frac{\|d\|_{2}^{2}}{2 ... 1answer 58 views ### Inequality with sum of rows of symmetric matrices Let S be a symetric matrix, with coefficients positive or zero, and T its square$$T=S^2 Let $S_i$ and $T_i$ be the sum of the $i$-th row (or column) of $S$ and $T$ respectively. I noticed that ...
How can a show the following assertion? If $(a_{ij})$ is a positive definite matrix and $\lambda,\Lambda$, the minimum and maximum eigenvalues, then ...