# Questions tagged [projection]

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412 questions
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### How to find Orthogonal Projections?

I am very confused regarding the topic orthogonal projections, so I will be really thankful if someone could help me. In my script is written that in order to find the orthogonal projection of a ...
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### Derivative of a function including Euclidean projection?

Let $C$ be a closed convex set in $\mathbb{R}^n$. Let $z$ be a point in $\mathbb{R}^n$. Definition: Euclidean projection of $z$ onto $C$ is defined as $$\pi_C(z)=\arg\min_{x\in C} \|z-x\|_2$$ ...
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### Inequality on pairs of projections in Kato's book

I do not understand an argument (p. 58, l.2--3) regarding two "close" projections, in the proof of Theorem I.6.34, pp. 56--58, Kato's book "Perturbation Theory for Linear Operators". The setting is ...
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### what's spectral axiom

I encounter a proposition in an article: For any $0\leq x\leq 1$ in a C*-algebra enjoying the spectral axiom, there are projections $(e_n)$ such that $$x=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n.$$ ...
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### Math notation to define the operator that extract component of a vector?

Say $\mathbf{x} \in \mathbb{R}^n$, what is the common notation to extract the first component as an operation? Something like $\mathcal{P}_j = \mathbf{x}_j$? $\mathbf{x}_j$ is the j-th component. I ...
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### Let $P ⊆ X ×Y$. Does $π_1(P)×π_2(P) = P$? Give a proof or a counterexample.

Proofs and fundamentals, exercise 4.2.5. I need your help, maybe it's false. Let $X$ and $Y$ be sets, let $A ⊆ X$ and $B ⊆ Y$ be subsets and let $π_1 : X × Y → X$ and $π_2 : X × Y → Y$ be projection ...
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### Matrix of a non-orthogonal projection and idempotence [closed]

Can find materials with the proof for LA question. So: Is this true that the matrix of a non-orthogonal projection is idempotent?
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### Projection map from projective space is not well defined?

Let $f: \mathbb{P^2} \to \mathbb{P^1}$ send $(x,y,z)$ to $(x,z)$. My book is telling me that this map is not well defined at $(0,0,1)$. How is this the case? Here $\mathbb{P}^1$ and $\mathbb{P}^2$ are ...
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### Is a von Neumann algebra a closed linear span of pairwise orthogonal projections?

It's well known that a von Neumann algebra is a closed linear span of its projections. Can we require these projections to be pairwise orthogonal? that is, can we find a set $\mathscr P$ ...
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### Least-Squares Opposite

Is there an opposite formulation of least-squares projection where the distance between each point–say, $(x_{i}, y_{i})$– and the subspace that it's projected onto (e.g. a line through the origin) is ...
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