# Questions tagged [matrix-norms]

This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

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### $\lVert P \rVert_2 =1$ iff $P$ is an orthogonal projector - Proof

I need help with understanding a step in a proof for the following exercise: Let $P\in\mathbb{C}^{m\times n}$ be a non-zero projector. Show that $\lVert P \rVert_2 =1$ iff $P$ is an orthogonal ...
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### The positive semidefinite(psd) of the 2 norm for vector

Suppose $q_n=(x_n,y_n)$ is a vector, $\Vert \cdot \Vert_2$ is the standard 2-norm like $\Vert q_n \Vert_2=\sqrt{x_n^2+y_n^2}$. Is $\sum_{n=0}^{N}\Vert q_{n+1}-q_{n} \Vert_2$ positive semidefinite(psd)...
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### Show that the set of extreme points of $S$ is its boundary

Let $S:=\{x:x^tx\leq 1\}$. Show that the set of extreme points of $S$ is its boundary. An extreme point, in mathematics, is a point in a convex set which does not lie in any open line segment joining ...
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### How to denote $L_p$ norm of a matrix $X$ in either row or column direction?

How to denote $L_p$ norm of a matrix $X$ in either row or column direction? The result of such an operation will be a column or row vector. Representing $\| X \|_p$ is ambiguous, isn't it? Do you have ...
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### Lower bound on the off-diagonal elements of a PSD matrix

Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, ...