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Questions tagged [projection-matrices]

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How do we derive the spectral projector associated with a simple eigenvalue?

Result 7.2.12 of Meyer's Matrix Analysis and Applied Linear Algebra gives the following: If $x$ and $y^*$ are respective right and left eigenvectors of a matrix $A$ associated with a simple ...
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solution projected on a subspace, reference request

$X_{n+1}=AX_{n}+B_n$ Where $B_n$ is sequence of i.i.d random variable defined in suitable spaces. $X_n$ are $\mathbb R^n$ valued random variable, $A\in M_n(\mathbb R)$ is column stochastic, $B_n$ are $...
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Given $\textbf{P} = \textbf{X}(\textbf{X}^{\prime}\textbf{X})^{-1}\textbf{X}^{\prime}$, prove that $\mathcal{C}(\textbf{P}) = \mathcal{C}(\textbf{X})$

If $\textbf{X}\in\textbf{R}^{n\times p}$ has full rank ($n\geq p$), so that $\textbf{P} = \textbf{X}(\textbf{X}^{\prime}\textbf{X})^{-1}\textbf{X}^{\prime}$, prove that $\mathcal{C}(\textbf{P}) = \...
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1answer
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difference of two orthogonal projections is orthogonal projection

Premise: I have an $n × q$ matrix $X$ and a $q × a$ matrix $C$ with $n > q > a$. I'm interested in the structure of the matrix $$ M = X X^+ - X_0 X_0^+ $$ where the superscript $^+$ indicates ...
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Projection matrix onto a subspace parallel to a complementary subspace

Given an $n \times k$ matrix $A$ and an $n \times (n-k)$ matrix $B$ such that $\text{span}(A) \oplus \text{span}(B) = \mathbb{R}^n$, how to get the projection matrix on $\text{span}(A)$ parallel to $\...
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Show that $||v||^2 = ||P_0v||^2 + ||v - P_0v||^2$ for orthogoonal projection

I'm working on some practice problems from Noble & Daniel's Applied Linear Algebra (3rd), specifically here looking for help with question 5 from section 5.8 on pg. 232. Suppose that $P_0$ is the ...
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1answer
24 views

Kernel of orthogonal projection on an eigenspace

let $Q$ be a $d\times d $-matrix and $P:\mathbb{R}^d \to \mathbb{R}^d$ be the orthogonal projection on the eigenspace $E_0 $ of $Q$. Why is the kernel of the projection the sum of the other ...
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Projecting a selection of points from a regular 2D grid onto a line

I would like to: start with a regular 2D grid like shown in the picture chose a line of slope $\tan \alpha$, and project, and a window of acceptance (grey area) project the points of the grid within ...
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1answer
30 views

Let $A$ be an operator, such that $A^{\dagger}A$ is a projector. Show that $AA^{\dagger}$ is also a projector

There is a hint to this problem which I don't know how to interpret. The hint is $$ \text{Hint: Show that} \quad A | \phi \rangle = 0 \leftrightarrow A^{\dagger}A=0. $$ Attempted solution(without ...
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1answer
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Intuition behind finding projection matrices of transformation

Supposedly a matrix $A$ is diagonalisable iff it can be written $A=\sum \lambda_i \pi_i$ where $\lambda_i$ are the distinct eigenvalues of $A$, and $\pi_i$ are projections satisfying $i \neq j \...
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Orthogonal projection of a vector on a linear subspace

I know this question has already been asked before but I'm not quite sure I've understood them well enough. I want to find the projection $\vec{v}_{proj}$ of $\vec{v}\in V$ on the span of the vectors ...
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1answer
44 views

Determine if a matrix is an orthogonal projection matrix

We define an orthogonal projection as a linear transformation that maps a vector into its orthogonal projection in some (given ahead) subspace $W$. Let's call the matrix of that transformation (...
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1answer
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Difference between the projection matrices arising from 1) the normal equations, and 2) an orthonormal basis of a subspace.

I'm having a bit of trouble tying two related ideas together, and I think I'm just missing a silly detail somewhere. In the standard formulation of linear least-squares, it can be shown that given a ...
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2answers
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Why is $U ^tU $ an orthogonal projection on $\operatorname{Im}(U)$?

Let $U \in M_{n,k}(\mathbb{R})$ such that : $^t UU = I_k$. Then I would like to understand geometrically why $U ^t U$ is the orthogonal projection on $\operatorname{Im}(U)$ ? When $n = k$ we are ...
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Find the unit vector within a subspace with the minimum norm projection onto another subspace

Let $W$ and $V$ be subspaces of $\mathbb{R}^n$ with dimensions $m$ and $p$ respectively. I want to find the unit vector in $W$ whose projection onto $V$ has the minimum Euclidean norm. From geometric ...
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An orthogonal projection induced by an $m \times n$-matrix

I am reading "Interlacing Eigenvalues and Graphs" by Willem H. Haemers. Right at the start in the proof of theorem 2.1 there is a step (marked in a red box below) which I do not understand. My ...
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Basis of subspace defined by linear functional

Let $V=span\{ b_1, \ldots, b_n\}$ be a $n$-dimensional vector space. Defined on $V$ is a linear functional $\Lambda$. Let $$V_\Lambda:= \{x \in V: \Lambda(x) = 0 \}$$ How can I find a basis of $V_\...
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1answer
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When is the sum and difference of two projection matrices $P_1$ and $P_2$ a projection matrix?

Let $P_1$ and $P_2$ be two projection matrices for orthogonal projections onto $S_1 \in \mathcal{R}^m$ and $S_2 \in \mathcal{R}^m$, respectively. When does $P_1+P_2$ and $P_1-P_2$ result in a ...
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Matrix of an orthogonal projector represented in the canonical basis

Find the matrix (represented in the canonical base $\{(1,0,0),(0,1,0),(0,0,1)\}$) of the projector $P:\mathbb{C^3}\rightarrow\mathbb{C^3}$ onto the subspace $$M=[\{f_1,f_2\}]=[\{(0,0,1),(\frac{2}{\...
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Higher moments of linear regression residuals?

Background In the following linear regression with i.i.d $\epsilon_i$ $(i = 1, \cdots, n)$ with mean 0 finite variance $\sigma^2$, \begin{align*} Y_i = X_i^\intercal\beta + \epsilon_i \end{align*} ...
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Fisher's formalism - which conditions to get equivalence between Fisher matrix and inverse of covariance matrix

I am currently studying Fisher's formalism as part of parameter estimation. From this documentation : They that Fisher matrix is ​​the inverse matrix of the covariance matrix. Initially, one builds ...
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find a vector with almost equal projection onto multiple vector

I have reached to a problem and I appreciate any suggestion to solve this issue. I need to find a vector like $v\in \mathbb{C}^M$ which has almost same projection onto multiple vectors with the same ...
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1answer
27 views

How to find the near close aspect ratio of a billboard from a distance photo?

I have a photo of a billboard. I am trying to solve this problem of finding out the aspect ratio of this billboard. What is known to me is nothing more than this photo. Can someone help me with ...
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29 views

Is it oblique projection matrix $P=x.y*$

Let $A \in C^{n \times n}$ be nondefective matrix and let $x$ and $y$ be right and left eigenvectors of $A$, (with corresponding simple eigenvalue $l$). Then we have that $y^*.x=1$, where $y^*$ is the ...
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25 views

Projection of a vector into the nullspace of a matrix

I need a clarification about the correct way to compute the projection of a vector into the nullspace of a matrix. For sake of clarity, let's call $A$ the matrix, $N(A)$ it's kernel and $A^\sharp$ ...
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What is the set of all isometric matrix in $\mathbb{R}^{k \times d}$?

An isometry from metric space $X=\mathbb{R}^{d}$ to metric space $Y=\mathbb{R}^{k}$ with usual norm for both spaces is the following: $$ \Phi: \mathbb{R}^{d} \rightarrow \mathbb{R}^{k} $$ where $\Phi(...
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1answer
38 views

Eigenvalues of Orthogonal Projection, using representative matrix

Let $V$ an inner product vector space and $U$ a vector subspace of $V$. Consider the linear operator $Proj_{\; U }:V\rightarrow V$ such that $\forall v\in V \; : \; Proj_{\; U}(v) = Proj_{\; U}V$, ...
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1answer
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Identity involving a projection matrix, originating from statistical regression theory

I am studying multiple linear regression and I am not able to understand a passage from my statistics textbook. The part that I do not understand can be formulated entirely in the language of linear ...
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50 views

Projection Matrixes, $A,B$ such that $\text{Id} - (A+B)$ is invertible

This question was on an old qual exam and I have been stuck on it: Let $A,B$ be two real $5x5$ matrices such that $A^2=A , B^2 = B$ and $\text{Id} - (A+B)$ is invertible. Show Rank($A$)=Rank($B$). ...
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1answer
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If matrices $A$ and $AB$ have full column rank, how do I prove that $P_A - P_{AB}$ is positive semidefinite?

First of all, the projection matrix $P_A$ is given by $P_A = A(A'A)^{-1}A'$. Similarly, $P_{AB} = AB(B'A'AB)^{-1}B'A'$. I have tried proving that $P_A - P_{AB}$ is itself a projection matrix, then it ...
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2answers
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Inequality involving projectors on a Hilbert space isomorphic to $\mathbb{C}^n$

Suppose we have Hermitian operators $P_1$ and $P_2$ on Hilbert space $\mathbb{H} \cong \mathbb{C}^n$, such that: $P_1^2=P_1$, $P_2^2=P_2$, $\{P_1, P_2\} = 0$. Let $x$ be a unit $L_2$-norm vector. ...
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Random Projection algorithm is strictly not a projection?

Current implementations of the Random Projection algorithm reduce the dimensionality of data samples by mapping them from $\mathbb R^d$ to $\mathbb R^k$ using a $d\times k$ projection matrix $R$ whose ...
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1answer
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Let $(v_1, v_2)$ be a basis for $\text{Im}(p)$ and $(v_3)$ be a basis for $\ker(p)$. Prove that $B' = (v_1, v_2, v_3)$ is a basis for $V$ .

Let $B = (1, X, X^2)$ be a basis for $\mathbb{R}_2[X]$ and $p ∈ \mathcal{L}\big(\mathbb{R}_2[X]\big)$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$...
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1answer
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Prove that $p ◦ p = p$. (Representing a Linear Transformation as a Matrix)

Let $B = (1, X, X^2)$ be an ordered basis for $\mathbb{R}_2[X]$ and $p ∈ \mathcal{L}(\mathbb{R}_2[X])$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)...
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1answer
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How to find projection matrix of the singular matrix onto fundamental subspaces?

\begin{align} A & =\begin{bmatrix}2&3\\6&9\end{bmatrix} \end{align} I found four fundamental subspaces of A which are \begin{align} C(A) & =\begin{bmatrix}2\\6\end{bmatrix} , ...
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1answer
51 views

Projection matrices $\mathbf{A}^{+}\mathbf{A}$ and $\mathbf{A}\mathbf{A}^{+}$

We are learning about pseudoinverses using the Strang book and I am just confused as to how to interpret the pseudoinverse. How come $\mathbf{A}^{+}\mathbf{A}$ projects into row space and $\mathbf{A}...
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1answer
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Find the projection of V in the direction of vector W

v = [2,3] and w = [0,1] projw(v) = w*v/||w||^2 = 3[0,1]=[0,3] Did I do the right step? Thank you
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1answer
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Find a matrix of oblique projector

How I should find this oblique projector matrix in the following task? Information at Wikipedia seems to be a little bit complicated and I haven`t found any practical examples for oblique projections. ...
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Clarification of projection

So I'm confused about projections. Following from my question here What is the difference between this question $1)$ "Find the standard matrix of the linear operator $T:R^2\rightarrow R^2$ given by ...
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2answers
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Orthogonal projection onto a vector with matrix transformation

So I have this question here which says: $a)$ Find the standard matrix of the linear operator $T:R^2\rightarrow R^2$ given by the orthogonal projection onto the vector $(1,-2)$. $b)$ Given the ...
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1answer
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Equality of orthogonal projection norms. [closed]

Let $F,G$ be two sub vector spaces of $\mathbb{R}^n$, and denote $P_F,P_G$ the orthogonal projection matrices on $F$ and $G$ respectively. Suppose that for all $v \in \mathbb{R}^n$ we have: \begin{...
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1answer
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Prove one projection matrix is larger than another

Please give me some hints for the right direction, but don't give me a full answer. We define $\mathbf{P_X}$ as projection matrices: $\mathbf{P_X = X(X'X)^{-1}X'}$. My exercise reads: Prove if both $...
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How to get from $I-p p^T$ to $-E p p^T E$?

$p \in \mathbb{R}^2$ is a unit-length column vector. $E=\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ is the "$-\frac{\pi}{2}$" rotation matrix. So how to prove that \begin{equation} I-...
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When is the nullspace unique in this case?

I am given the following: $$ I - AA^T $$ is a projection matrix onto the orthogonal complement of $< A >$. So the nullspace of $I-AA^T$ is the subspace spanned by the set of vectors $x$ such ...
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Consider the plane P in R-3 given by x-y-2z=0

I found the matrix A whose columns are a basis for P, A=[1,-1,-2] (vertical form). Using that I was able to find the projection matrix: P=$\frac{-1}{2} \left( \begin{array}{cc} 1 & -1 & 2 \\...
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Proof of equality of OLS projection matrix and GLS projection matrix

I'm struggling with the proof of the following proposition: Given a $n\times n$ symmetric, positive-semidefinite matrix $\Omega$, a $n\times k$ matrix $X$ such that $rank(X)=k$, and an invertible ...
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118 views

Define on $P3$ the inner product $<f,g>=\int_{-1}^1 f(t)g(t)dt$, find orthogonal projection

Define on $P3$ the inner product $\langle f,g \rangle=\int_{-1}^1 f(t)g(t)dt$. a) find the orthogonal projection of $p(x)=x^3$ onto $P2$ I know the orthogonal projection formula, but how do I solve ...
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23 views

Parametrization and Projections

I have a question over this problem that I've encountered. I don't know how to solve it. Let $\ell$ be the line parametrized as $(t, 2t+1, 3t+2)$ and let $P$ be the plane with equation $x+y+z = 1.$ (...
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1answer
43 views

What does it mean by a matrix is a projector on some subspace?

Could anyone explain to me what does mathematically mean a matrix $A$ is a projector on some subspace $V\subset\mathbb R^n$ in general? Does it mean $(A^2-A)x=0\forall x\in V$? Thankx.
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1answer
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perspective projection transformation matrix

In one text, the derivation for perspective projection goes like this: If $x',y',z'$ represent any point along the projection line,and $x_{prp},y_{prp},z_{prp}$ are the projection ...