# Tagged Questions

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### Relation between determinant and L1 norm

Recently, I have coped with a problem about the relation between determinant of positive definite matrices and their L1 norm. More specifically, assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two ...
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### 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 ...
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### Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
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### 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). ...
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### Prove relative error with condition number of matrix inequality

I was working on some questions and solutions, and encountered the following question. I am able to prove the inequality using the given information and some algebraic manipulation but the "within ...
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### $A'B=I \rightarrow B'B \geq (A'A)^{-1}$

Could anyone please help me show that for general matrices it holds that $$A'B=I \rightarrow B'B \geq (A'A)^{-1}$$ Thanks.
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### 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 ...
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### How to prove this matrix bound

Let an $m$ by $n$ matrix $A\in\mathbb C^{m\times n}$. Denote $M=\max_i\sum_{j=1}^n|A_{ij}|$ and $N=\max_j\sum_{i=1}^m|A_{ij}|$. Prove for any two vectors $x\in\mathbb C^m$ and $y\in\mathbb C^n$, we ...
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### 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. $x_i \geq 0$ $\forall i$. ...
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### bound on trace-norm of product of matrices

Is it true that $$\|ABA^\dagger\|_1\leq \|A\|^2\|B\|_1,$$ where $\|A\|_1$ is the trace norm, $\|A\|$ is the spectral norm, and $A$ and $B$ are square matrices?
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### Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
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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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### Eigenvalues of sum of Hermitian matrices

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### How prove this matrix inequality $\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$

Question: Let $A_{n\times n},B_{n\times n}$ be positive Hermitian matrices. 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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### Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$
I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...