All Questions
Tagged with projection orthogonality
93 questions
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SageMath: Orthogonal projection of $\mathbb{C}^3$ onto a subspace.
Consider the operator $T(x,y,z)=(x-iy+iz,ix-z,2y).$ I want to compute the orthogonal projection of $\mathbb{C}^3$ onto im$(T)$ using the projection formula $$P(v)=\langle v,w_1\rangle w_1+...+\langle ...
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$UU^*$ by columns
Background
Let $U\in \mathbb(C)^{n \times m}$ with $m<n$ have the columns $u_k$. Namely,
$$U = \begin{pmatrix}
\vert & & \vert \\
u_1 & ... & u_m \\
\vert & &...
- 381
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Lipschitz constant of projected function
Suppose $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ is Lipschitz with constant $K$. Suppose I project $f$ onto orthogonal subspaces of $\mathbb{R}^n$, say $X$ and $Y$, where $\mathbb{R}^n= X \oplus Y$ ...
- 1,142
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Characteristic polynomial of an orthogonal projection
Q: What is the characteristic polynomial of an orthogonal projection onto a (two-dimensional) plane through the origin in $\mathbb R^4$?
Ans: $x^2(x-1)^2$
Can someone please explain how to do this ...
- 629
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61
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Further decomposition of isotypic components in a representation
Let $(V,\rho)$ be an orthogonal (resp. unitary) representation of finite group $G$ whose irreducible representations over the same field as $V$ are $W_i$ with character $\chi_i$.
We have $V \cong \...
- 2,596
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25
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Projection of vectors from starting basis onto orthogonal complement
Gram-Schmidt process allows us to produce a basis $\{w_1,...,w_n\}$ starting from a basis $\{v_1,...,v_n\}$. If I define $W_j$ to be the subspace generated by $\{w_1,...,w_j\}$ for $j=1,..,n.$ Can I ...
- 1,392
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Linear Algebra: Orthogonal basis to find proj of $\mathbf{w}$ onto $\mathbf{y}$?
I'm currently studying for my final exam for linear algebra, and I'm a bit confused about how to find the projection of $\mathbf{w}$ onto $\mathbf{y}$.
I already found the orthogonal basis for $W$, ...
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Prove P²=P for orthogonal projection formula
Given the following orthogonal projection formula $\hat{x}_w$ of $x \in \mathbb{R}^n$ on the vector $\frac{w}{\|w\|} \in \mathbb{R}^n$ defined by $\hat{x}_w := \frac{c \cdot w}{\|w\|}$ where $c = \...
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Shortest distance between two affine subspaces through orthogonal projection
I'm trying to show the following:
Let $V$ be a finite dimensional euclidean vector space, with two vector subspaces $S_1 \subset V$ and $S_2 \subset V$. Suppose that $X, Y$ are affine subspaces with $...
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Is $\langle f - m, m \rangle = 0$ sufficient for $m$ to be an orthogonal projection?
I was looking at https://en.wikipedia.org/wiki/Hilbert_projection_theorem and I wondered about the following statement, but I wasn't able to prove or disprove it.
Suppose that $M$ is a subspace of a ...
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What do the parts of the Gram-Schmidt process mean and represent in space?
I am struggling to understand what the different parts of the Gram-Schmidt process represent.
Suppose we have a basis $\{x_1, x_2\}$
We would then find a orthogonal basis by doing the following :
$$...
- 11
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Finding an explicit formula for the projection on a subspace of an Hilbert space
Given the Hilbert space $L^2([-1,1])$ endowed with the usual inner product, consider the following operator:
$$Tf:=\int_{-1}^1f(x)e^x \mathrm dx $$
Let $N:=\ker T$, find an explicit formula to find ...
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Property of Orthogonal Projections
Let $H$ be a real Hilbert space with inner product $(\cdot, \cdot)$. Assume that there is a subset $C \subset H$ that satisfies
$$C=\{x \in H : (x,y) \geq 0\} \forall y \in C$$
$C$ can be shown to be ...
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Orthogonal projections and Orthogonal Complements
I'm reading on orthogonal projections from a course's notes and it says the following:
For each x ∈ $R_n$ and each linear subspace U, $\pi_U$(x) exists and is unique. Moreover, $\pi_U$(x) is the only ...
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self-adjoint projection
for the following exercise i have some questions, i appreciate some help or hints for this exercise.
Let be V an euclidean or unitary vectorspace and $ p:V \to V$ a self-adjoint Projection.
$(a)$ Show ...
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Question on showing that $\frac{1}{n}\sum_{k=0}^{n-1}U^kf$ converges in norm to the orthogonal projection $Pf$ to the space $\{f\in H: Uf = f\}$
Edit: The reference I am reading is Yves Coudène's Ergodic Theory and Dynamical Systems, chapter 1, proof of theorem 1.1 on page 5.
Let $H$ be a Hilbert space and $U:H\to H$ a bounded linear operator ...
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How to find an orthogonal transformation between two matrices [closed]
I wonder if there is a way to solve this problem:$$\arg \min_{\alpha, U} \|A - \alpha UB\|_\mathcal F^2 \\ \text{s.t. } \alpha \in \mathbb R, U \in \mathbb R^{n \times n} \text{ and } U^\top U = I$$
...
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Given orthogonal projections P and Q on a Hilbert space such that norm of P-Q is less than one. Then rank of P and Q w.r.t. Hilbert dimension same.
Given two orthogonal projections P and Q on a Hilbert space such that $\|P-Q\|<1$. Then dim(range(P))=dim(range(Q)) w.r.t. Hilbert dimension.
Please note that the definition of Hilbert dimension ...
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Finding an explicit formula for the orthogonal projection to the span of a single vector in a Hilbert space $H$
Let $H$ be a Hilbert space over the field $\mathbb{K}$ and $C$ a closed vector subspace of $H$. My reference defines the orthogonal projection of $u \in H$ onto $C$ as the unique element $P(u) \in C: |...
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Bounding $L^2$ norm of projection operator by inner product with arbitrary vector in projected subspace
Let $W$ be a subspace of a Hilbert space $H$ and let $P$ be a self-adjoint orthogonal projection operator onto $W$. Let $v \in H$. Then is it true that for any $w \in W$ with $||w||_2 = 1$, we have
$$...
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Exercise on Orthogonal Projection
For an exercise, I need to investigate the orthogonal projection in more detail. I know that this has probably been discussed many times here, I am though interested in whether my explanations suffice ...
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Orthogonal project of a vector $\vec{y}$ that sit in the same plane (aka span($u_1, u_2$)) such that $\vec{y},\vec{u_1}, \vec{u_2} \in \mathbb{R^3}$
Confirmed that $u_1,u_2$ are indeed orthogonal by taking their dot product which equals zero. Correct answer = $\begin{pmatrix}-1\\ -1\\ 6\end{pmatrix}$.
Geographically, mapped out in geogebra: we see ...
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195
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orthogonal projection of irreducible representation
Let $G$ be a finite group, K a subgroup, and $X = \frac{G}{K}$ the space of cosets. Consider $L(X) = \bigoplus_{m=0}^N V_i$ the decomposition of $L(X)$ into irreducible sub-representations, in ...
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Why can we reformulate $ \lvert \lvert Y-P_{[X]}Y\rvert \rvert ^{2}-\lvert \lvert Y-P_{[X_{0}]}Y\rvert \rvert ^{2}$ in the following way?
Let $[X]$ and $[X_{0}]$ denote the span of $X$ and $X_{0}$, respectively. Further, let $P_{[X_{0}]}$ denote the orthogonal projection onto the subspace $[X_{0}]$, and $P_{[X]}$ denote the orthogonal ...
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$\text{ran}\,N$ is closed if and only if $0$ is no limit point of $\sigma(N)$, for a normal bounded operator $N$ on a hilbert space $H$
This question has been asked once in this forum, but I don't understand some things in the proof of one direction. I marked them in bold:
${\Longleftarrow}:$ Let $\lambda\in\sigma(N) $ be an ...
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Suppose $P$ and $Q$ are orthogonal projections and $P+Q = I$. Prove that $P-Q$ must be an orthogonal matrix.
By theorem of orthogonal projections we have $P^T=P=P^2$ and $Q^T=Q=Q^2$. By definition we need to show that $(P-Q)^T(P-Q) = I = (P-Q)(P-Q)^T$. Note that $P+Q = I \iff P = I - Q$. Observe
\begin{...
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If $P$ is an orthogonal projection, prove that $P^+ = P$.
If $P$ is an orthogonal projection, prove that $P^+ = P$. Where '$^+$' indicates the Moore-Penrose pseudo-inverse.
What we know is the $P$ is symmetric and idempotent. That is $P = P^2 = P^T$. I am ...
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Functions orthogonal to linear span when considering a normal distribution
Let $x \sim N(0, 1)$ be a random variable distributed as standard normal.
I am looking for functions that satisfy $$E[x f(x)] = 0 $$.
In particular, functions whose best approximation is a constant ...
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What is the orthogonal part?
I have heard multiple explanations for what makes something an ORTHOGONAL projection and it's quite confusing.
The first explanation of orthogonal is that if you project b on C(A), the nature of ...
user946705
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Is this a mistake in projection formula?
I am referencing Chad's answer from this link:
https://math.stackexchange.com/a/2997916/
I have also added an image from the referenced link:
enter image description here
If b is being projected.....
user946705
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Norms of orthogonal subspaces
Let V be a finite dimensional real inner product space and U, W subspaces of
V such that U is orthogonal to W. Show that for any $v ∈ V$
$$||v||^2 ≥ ||proj U (v)||^2 + ||projW (v)||^2$$
Hello guys. I ...
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Projection onto a Line: why must the projection be a multiple of the vector $a$?
In the book: Linear Algebra and its application -Gilbert Strang-, when the projection $p$ of vector $b$ onto vector $a$ is explained, this is said:
We want to find the projection point $p$. This point ...
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Let $P$ be an idempotent linear operator on V. Then if $\text{null}(P)\subseteq(\text{Im}(P))^\perp$, $P$ is an orthogonal projection.
I'm trying to understand why the statement above is true for a finite dimensional inner product space V.
After some research I found that as long as $P$ is a linear operator on V and is idempotent, ...
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Proof and meaning of x'' being minimal in orthogonal projections of the form x = x' + x''
I am reading the book Self-organizing maps by Kohonen.
In Theorem 1.1.1 he states
Of all decompositions of the form $x = x' + x''$ where $x'\in
\mathcal{L}$, the one into orthogonal projections has ...
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How to prove that $P\mathcal H \perp Q \mathcal H$ if $P,Q$ and $P + Q$ are all projections?
Let $\mathcal H$ be a Hilbert space and $P,Q \in B(\mathcal H)$ are projections. Then $(P+Q)$ is a projection iff $P \mathcal H \perp Q \mathcal H.$
"$\impliedby$" part is easy. How do I ...
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If $A$ is a bounded operator on a Banach space $X$, is there a topological decomposition $X=\mathcal N(A)\oplus_1X_2$?
Let $X,Y$ be Banach spaces and $A\in\mathfrak L(X,Y)$.
Under which conditions can we show that
$X=\underbrace{\mathcal N(A)}_{=:\:X_1}\oplus_1\underbrace{(1-\pi)X}_{=:\:X_2}$ (topological direct sum)...
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Orthogonal projection onto a subspace
Let $v_1=\begin{pmatrix}1\\ 1\\ 1\\ 0\end{pmatrix}$ and $v_2 = \begin{pmatrix}0\\ 1\\ 1\\ 1\end{pmatrix}$. Find the orthogonal projection of $v = \begin{pmatrix}3\\ 2\\ 1\\ 0\end{pmatrix}$ onto $V= \...
user854387
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Question about orthogonal projection and some properties
Let be $V$ a vector space with an inner product, $W \subset V$ a subspace with $\dim W < \infty$ and $P:V \to V$ a projection s.t $Im P = W$ and $\vert \vert P(v) \vert \vert \leq \vert \vert v \...
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Does projection operation preserve perpendicularity?
Let $C$ be a nonempty convex closed subset of $R^n$, and $\pi_C$ be an orthogonal projection mapping from $R^n$ onto $C$. Suppose that $y \in R^n$ is perpendicular to $x \in R^n$. Is it possible to ...
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Finding the orthogonal projection on a subspace $K \in \ell^2(\mathbb{N})$
I'm trying to find the orthogonal projection of $\ell^2(\mathbb{N})$ on $K$ and then determining $p_K(x)(2n)$ for $x\in \ell^2(\mathbb{N})$ and $n\in \mathbb{N}$. The subspace K is defined as
$$ K = \{...
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Show that $\{1, \cos x, \sin x, \cos(2x), \sin(2x), \dots, \cos(nx), \sin(nx)\}$ is an orthogonal set of V .
Suppose $V := CR([0, 2\pi])$ is the collection of real valued functions on $[0, 2\pi]$
with inner product given by
$$\langle f,g\rangle:= \int_0^{2\pi}f(x)g(x)\,dx$$
Show that $$S_n := \{1, \cos x, \...
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70
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Finding the projection of v onto the orthogonal complement of u
Let $X = \mathbb{R}^{n}$. Known there is an orthogonal basis $\{u, q_1, q_2,...,q_n\}$ for $X$, but only the first basis vector $u$ is given explicitly. Let $v$ be an arbitrary vector in $X$ with ...
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Is the orthogonal projection of y onto two orthogonal vectors the value of y?
Is the orthogonal projection of $y$ onto two orthogonal vectors always simply $y$?
I am doing some self-guided study of linear algebra, and something in the course seemed to imply this was the case ...
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Rudin's Real and Complex Analysis, Section 9.16
In Section of 9.16 from Rudin's RCA, it says
Let $\hat{M}$ be the image of a closed translation-invariant subspace $M \subset L^2$, nder the Fourier transfrom. Let $P$ be the orthogonal projection of ...
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52
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Project orthonormal vectors onto subspace while preserving orthonormality
Suppose I have $m$ orthonormal vectors $u_1, ..., u_m \in \mathbb{R}^n$ where $m < n$. I would like to project each vector onto $\mathbb{R}^m$ using some linear transformation $W: \mathbb{R}^n \...
- 1,113
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Passage missing in simple proof of orthogonal decomposition
I am reading this nice book on linear algebra. Specifically, I am reading the proof of the Theorem for Orthogonal decomposition of a vector $x\in \mathbb{R}^n$, given a subspace $W$. I think there is ...
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297
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Minimizing the distance to a subspace (orthogonal projection) if a norm is not induced by an inner product
Let $V = \mathbb{R}^n$ with the inner product $\langle\cdot,\cdot\rangle$ and $U \subset V$ a vector subspace. Then for $v\in V$ and $x \in U$ the inequality$$
||v-x|| \leq ||v-u||$$ is satisfied for ...
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Range of a unitary transformed orthogonal projection
Let $X$ be a Hilbert space, $P$ an orthogonal projection in $X$ and $Q \in L(X)$ (i.e. $Q \colon X \to X$ is linear and continuous)
a unitary linear transformation, i.e. $Q^*Q=Id_X= QQ^*$ ($Q^*$ ...
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Given a family of $n$ vectors, each of distance $1$ from the span of the rest. Is the norm of their sum at least $\sqrt{n}$?
Let $v_1,\dots,v_n$ be vectors in $\mathbb{R}^n$, not necessarily linearly independent. For each $v_i$ let $v_i'$ denote the projection of $v_i$ into the orthogonal complement of $\mathrm{span}\{v_1,\...
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Question regarding orthogonal complement.
We know that for any subspace $W$ of an inner product space $V$,we have $W \subset {W^\perp}^{\perp}$.We also know that for $W$ finite dimensional $W={W^\perp}^\perp$.Now I think the equality will ...
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