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Questions tagged [projection-matrices]

This tag is for questions relating to projection matrix, which is an square matrix that gives a vector space projection from to a subspace.

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projections from a plane in $\mathbb{R}^6$ onto three orthogonal planes

Let $\Pi_1=\operatorname{span}\{e_1,e_2\},\Pi_2=\operatorname{span}\{e_3,e_4\},\Pi_3=\operatorname{span}\{e_5,e_6\}$ be orthogonal 2D planes in $\mathbb{R}^6$. Let $U$ be arbitrary 2D plane in $\...
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Link between summing orthogonal projections and Projection Matrix $A(A^TA)^{-1}A^T$

Assuming the usual inner product $\langle x, y\rangle = x^\mathsf{T} y$ on a real vector space $V$, I believe we can define the orthogonal projection $\mathbf P_W\colon V\to W$ as $$\mathbf P_W(v) = \...
CormJack's user avatar
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Rank of products of projections

We consider two $n \times n$ projections, $A$ and $B$. In particular, this means that $A^2 = A$ and $B^2 = B$. Given this, I was curious on if products of projection matrices would have the same rank? ...
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Veryshynin HDP Exercise 4.1.4: Equivalent definitions of isometry

I'm trying to show the following: Let $A$ be an $m \times n$ matrix with $m\geq n$. Prove that the following statements are equivalent. $A^{\top} A = I_n$. $P = A A^{\top} $ is an orthogonal ...
pbb's user avatar
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Understanding how matrix transformations operate across subspaces (e.g. rowspace, columnspace)

My question originates from a short proof in Strang's LA book that shows when $A\pmb{x}=\pmb{b}$ every vecetor $\pmb{x_r}$ in the rowspace of $A$ projects uniquely to a single vector $\pmb{b_c}$ in ...
Joseph's user avatar
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Find the projection of a vector onto a subspace along the orthogonal complement of another subspace

what I want to prove is that if $P \in \mathbb{R}^{n \times n}$ is a projection matrix onto $\mathbb{S}_1$ along $\mathbb{S}_{2}^{\perp}$ , $v_1, v_2, \ldots, v_m$ and $w_1, w_2, \ldots, w_m$ ...
YuerCauchy's user avatar
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Construct perspective projection of rotating tesseract by perpendicular lines intersecting ellipse

The contruction was used in two different sources on the web: a Geogebra resource and a video using inRm3D so I think it must be documented and proved somewhere, but I didn't find any. Here is the ...
hbghlyj's user avatar
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Johnson Lindenstrauss lemma for positive entries matrix

I read about the Johnson Lindenstrauss lemma, which (roughly) states that if $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{k \times n}$ is a "random matrix", then with high probability $\|x\|...
Johana T's user avatar
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"Agreeing" orthogonal projections

This is a projection excercise I'm stuck on. Lets say we have two planes in R3, $A$ and $B$, which both go through the origin. We also have two orthogonal projections, $ProjA$ and $ProjB$, projecting ...
hellothere's user avatar
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Proving product of two projection matrices is commutative

I’ve been trying to understand the details of orthogonal projections within the context of linear regression models, and I’ve come across the following scenario: Given a linear regression model $Y=\...
Noy's user avatar
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Conditional distribution of random orthogonal projection matrix

I have encountered a rather curious question. Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n-$dimensional ...
Landon Carter's user avatar
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1 answer
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Distribution theory for random projections

Suppose $v$ is a fixed vector in $\mathbb R^n$, and let $u\in S^{n-1}$ (unit sphere in $n$ dimensions; $S^{n-1}=\{x\in\mathbb R^n:\|x\|=1\}$) be uniformly generated. What is the distribution of $\...
Landon Carter's user avatar
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Properties of a Product of Commuting Projections

Let $\Pi_1$ and $\Pi_2$ be two positive-semidefinite projections which commute, and let $\Pi=\Pi_1\Pi_2=\Pi_2\Pi_1$. Are the following statements correct? $\quad\Pi\:\preceq\:\Pi_1\,$ and $\,\Pi\:\...
Nick Cooper's user avatar
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What causes a trace of a product of two projection operators to not be a whole number

Generally it should mean that their product is not a projector itself(correct me if I'm wrong). But is the trace in this case indicative of something else? What does this mean intuitively? Also, if ...
BelguardianMPh's user avatar
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Given a $n \times n$ matrix $A$, when do the first $p$ columns of $A^{-1}$ coincide with the Moore–Penrose inverse of the first $p$ rows of $A$?

I noticed that for ${A}= \left[\begin{matrix} \cos{\left(\phi\right)} & \cos{\left(\phi+2\pi/3\right)} & \cos{\left(\phi+4\pi/3\right)}\\ \sin{\left(\phi\right)} & \...
Fabio Dalla Libera's user avatar
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Reducing the size of the product of a matrix and a vector

I have a matrix $P$ with size $(d,d)$ and two vectors $x$ and $y$ of size $(d,1)$. I want to reduce the size of the quantity $(Px-Py)$ by setting the size of $P$ to be $(p,d)$ where $p \leq d$ to ...
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Vector having same absolute correlation with all columns

Let $\mathbf{\Phi}\in \mathbb{R}^{m\times n}$ be a real valued matrix with $m\le n$. I want to find a vector $\mathbf{y}\in \mathbb{R}^m$ such that $\frac{\left|\langle \mathbf{y},\mathbf{\phi}_i\...
Samrat Mukhopadhyay's user avatar
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$\text{rank}(A)+\text{rank}(A-I)=n \iff A^2=A$. I don't know my mistake in my proof

Let $A$ be an $n\times n$ matrix over a field $F$. Show that $A^2=A$ if and only if $\text{rank}(A)+\text{rank}(A-I)=n$. My attempt: $A^2=A \implies \text{rank}(A)+\text{rank}(A-I)=n$ Using the ...
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Custom block-wise matrix

I have two block matrices $A$ and $B$ defined as: \begin{align} A = \begin{pmatrix} A_{11} & A_{12} & A_{13} & A_{14} \\ A_{21} & A_{22} & A_{23} & A_{24} \\ A_{31} & A_{...
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Does Matrix projection impacts centralization

I have a dataset that each sample is a matrix $A_{m\times n}$, and I have N samples in my dataset. I want to centralize my data across my samples, meaning that for each element in the matrix, I want ...
Madi's user avatar
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Eigenvalue relation of a symmetric matrix $A$ and $A + vv^T$

My question pertains to the material in the book "The Algebraic Eigenvalue Problem" by J.H. Wilkinson. Section "Symmetric matrix of rank unity", pages 96-97. The setup is as ...
MonteNero's user avatar
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von Neumann entropy in a reduced state space

I'm winding my head around a quite general physics issue for some time. Given some density operator (or its matrix representation) $\rho$, what happens to its von Neumann entropy, when some of the ...
Lars Hanke's user avatar
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239 views

How to determine if the orthogonal projection is onto a line or a plane?

I'm trying to determine if the projection using the $P$ matrix is onto a line or a plane, if the matrix is given as: $$P = \frac{1}{3} \cdot \pmatrix{2 & 1 & 1 \\ 1 & 2 & -1 \\ 1 & ...
Aleksa Majkic's user avatar
2 votes
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Rank of a matrix multiplied by random projections

Say we have a non-random matrix $\textbf{K}\in\mathbb{R}^{p\times p}$ with $\text{rank}(\textbf{K})=\rho\leq p$ and a random projection $\textbf{R}\in\mathbb{R}^{d\times p}$ with i.i.d. standard ...
Undertherainbow's user avatar
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Lower bound on smallest principal angle between two complementary subspaces

Let $U, V \subset \mathbb{R}^n$ be two subspaces spanned by the columns of $A\in\mathbb{R}^{n\times m}$ and $B = A + E$, respectively. Moreover, let $n = 2m$, $A$ and $B$ have full rank, and $U \cap V ...
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Reading off probabilities for measurement outcome rather than using projection operator?

Let $\alpha_{0}$ = $\alpha_{1}$ = $\frac{1}{\sqrt{2}}$. Suppose the state vector $| \psi \rangle = \alpha_{0}| \psi_{0} \rangle + \alpha_{1} |\psi_{1}\rangle $ describes a quantum mechanical system ...
Mathematicing's user avatar
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inner product (Bra - Ket) involving projection operator

In quantum mechanics, the action of a projection operator $\hat{P}$ acting on a quantum mechanical system, prepared in a state $| \psi \rangle$, is described by the eigenvector equation $\hat{P} | \...
Mathematicing's user avatar
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Interpretation/terminology of a matrix

I have two vectors ${\bf x} \in \mathbb{R}^d$ and ${\bf y} \in \mathbb{R}^p$ such that $d < p$ and let ${\bf A} \in \mathbb{R}^{s\times d}$ and ${\bf B} \in \mathbb{R}^{s\times p}$two matrices, ...
Chao's user avatar
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Proof of projection matrix multiplication with any vector

Im studying Projection matrix (linear algebra) and I understood that projection matrix onto line a can be calculated as: $\cfrac{a a^T}{a^T a}$ I know that multiplying any vector to this matrix ...
COTHE's user avatar
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Inverse and Projection matrices

Consider the following example first, we have matrix given as,($v\in\mathbb{R}^n$) $$a_{ij}=\delta_{ij}v^4+2v^2v_iv_j$$ Now, define the following projection matrix, $$I^{||}_{ij}=\frac{v_iv_j}{v\cdot ...
Swastik Majumder's user avatar
3 votes
3 answers
110 views

Standard Matrix

During an exam, we were asked to determine the standard matrix of the linear image, $P: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ which projects a vector onto the $y=-x$ axis. What I did is, I took the ...
Masterrun80's user avatar
3 votes
1 answer
67 views

Getting the original matrix from the projection

If I have the projection matrix of $X$, $$ P = X\,{(X^TX)}^{-1} X^T, $$ how can I recover $X$ by only knowing $P$? Is there a way to do that?
Madi's user avatar
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(idempotent)Does $rank(A) + rank(A-I_n)=n$ implies $A^2=A$? [duplicate]

Let $A\in M_{n\times n}(R).$ If $rank(A) +rank(A-I_{n})=n$, show that $trace(A)=rank(A)$ I have already known that an idempotent $A^2=A$ implies...$$(1)rank(A) +rank(A-I_{n})=n\qquad(2)trace(A)=rank(...
mlrofcloud's user avatar
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What can be said about $AA^+$ for symmetric real matrices?

Defining $A^+$ as the pseudoinverse of matrix $A$, what can be said about $AA^+$ if $A$ is a real matrix? What if $A$ is also symmetric? The reason I am asking this is that I want to see why the ...
HappyFace's user avatar
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Spectrum of the projection of a matrix

Let $A$ be a strictly positive symmetric matrix in $\mathbb{R}^{n \times n}$ and $P$ be an orthogonal projection onto a subspace $U$ of dimension $m \leq n$. Is there any way to relate the eigenvalues ...
DimSum's user avatar
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Is there a canonical way to express the orthogonal projection onto a hyperplane parametrized by a normal vector?

Suppose I have the hyperplane in $\mathbb{R}^p$ described by the equation $$\beta_1 x_1 + \cdots + \beta_p x_p = 0.$$ This hyperplane can also be thought of as the orthogonal complement of the set ...
TheProofIsTrivium's user avatar
1 vote
1 answer
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Projection matrix product under operator norm

I've come across a question that is similar to this one, but somewhat "reversed" in the role of the unitary matrix. Suppose I have matrix product $A\, P\, B$ where $A$, $B$ are generic real ...
Ocker's user avatar
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3 votes
2 answers
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Prove that a collection of certain polynomials sum to unity

Consider a (finite) collection of $n$ distinct constants $\{f_1,\dots,f_n\}$ and define the polynomial in $x$ $$P_i(x):=\prod_{j\in\{1,\dots,n\}}^{j\neq i}\frac{(x-f_j)}{(f_i-f_j)}~,$$ for $1\leq i \...
SigmaAlpha's user avatar
2 votes
1 answer
59 views

Finding a relation between $P$ and $P^2$

I encountered a challenging problem on a mock test today: Let $x$ be an $n \times 1$ matrix. We define $$P = -(x^Tx)^{-1}\cdot(xx^T)$$ Now there were $4$ options, something like $P^2 - P = O$ where $O$...
Nikunj's user avatar
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$A: R^{n} \rightarrow R^{n}$, $A^{3}$ - projection. Eigenvalues and diag matrix in any basis.

Here is task: $$A: R^{n} \rightarrow R^{n}$$ $A^{3}$ - projection. (a) - What eigenvalues this linear operator have? (b) - Is it true that A will have a diagonal matrix in some basis $R^{n}$? =========...
replikeit's user avatar
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Magic Bases/Magic Unitaries with rank one projections

Consider a matrix $a\in M_n(M_n(\mathbb{C}))$ such that each entry is an orthogonal projection $a_{ij}=a_{ij}^2=a_{ij}$ and the sum along any row or column is the identity: $$\sum_{k=1}^n a_{ik}=I_n=\...
JP McCarthy's user avatar
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4 votes
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Projection of a 3D circle onto a 2D camera image

Assume that I have a 3D circle with a center at $(c_1, c_2, c_3)$ in the circle coordinate frame $C$. The radius of the circle is $r$, and there is a unit vector $(v_1, v_2, v_3)$ (also in coordinate ...
BeginnersMindTruly's user avatar
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2 answers
179 views

Are there non-identity matrices that fit the definition of $A^2 = A$?

Consider the 'pseudo' definitions below: $A$ is an $n \times n$ matrix $A$ is $m_1$ if $A^2= I$ (the identity matrix) $A$ is $m_2$ if $A^2 = A$ Currently, the only type of matrix that is $m_2$ that I ...
testcase0_'s user avatar
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What is the correct way to connect the subspaces of a matrix to transpose vectors in the method of least squares?

I'm studying linear algebra alone and having a hard time dealing with transpose vectors and their subspaces. The summary of what I saw is as follows: The row space of $A$ is the column space of $A^T$ ...
ististyle's user avatar
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Prove orthogonal projections for a matrix A and its transpose

I'm trying to prove that a projection is orthogonal using: $A ∈ \mathbb R^{m\times n} $ where $A^TA = I_n$, and I want to prove that $P=AA^T$ is an orthogonal projection. I understand that an instance ...
kdata23's user avatar
1 vote
1 answer
98 views

Decompose the identity matrix into several projection matrices. Prove the range of $P_j$ is the subset of the kernel of $P_i$

Decompose a $N \times N$ identity matrix into $N$ projection matrices. $$ I = \sum_{1 \le i \le N} P_i \\ \forall P_i,\ P_i^2 = P_i $$ Prove $$\operatorname{Range}(P_j) \subset \operatorname{Kernel}(...
gyro's user avatar
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Projection operator (redundant?) definition

In his Quantum Mechanics, Ballentine says that “in general, a self-adjoint operator which obeys $\rho^2=\rho$ is a projection operator”. I'm not sure I follow the need for the self-adjoint caveat? I ...
EE18's user avatar
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Geometrically, why do we need independent columns in a matrix $A$ when computing the projection matrix onto the column space of $A$?

Consider an attempt to find the line $f(t)=C+Gt+Ht$ that best approximates a set of points using least squares. This is a contrived example to try to explain what exactly goes wrong when we have non-...
xoux's user avatar
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1 vote
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How to find the basis of a transformation matrix?

I need help with b. Lets call the column vectors of the transformation matrix $w_1, w_2, w_3$. I can already see that $w_3 = \begin{bmatrix} 1\\ 2\\ 2 \end{bmatrix}$ or simply the norm. But I am ...
Need_MathHelp's user avatar
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1 answer
213 views

Canonical form of projection matrices

I am reading through this paper by Stewart on oblique projectors, i.e. matrices $P \in \mathbb{C}^{n \times n}$ where $P^2 = P$. He describes a canonical form for projectors as follows. Let $P$ be be ...
Srinivas Eswar's user avatar

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