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Questions tagged [orthogonality]

This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

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Are there degrees or categories of orthogonality?

The vectors $\begin{bmatrix}a \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix}0 \\ b \\ 0\end{bmatrix}$, and $\begin{bmatrix}0 \\ 0 \\ c\end{bmatrix}$ are orthogonal under the dot product. The vectors $1$, $...
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Norm of the sum of orthogonal projections

Let $H_\lambda$ be a closed subspace of a $\mathbb R$-Hilbert space $H$ for $\lambda\ge0$ and assume that $(H_\lambda)_{\lambda\ge0}$ is nondecreasing and right-continuous, i.e. $$\bigcap_{\mu>\...
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Invariant Subspaces and Orthogonality

I'm struggling with a question on invariant subspaces. If someone could possibly help me out here that would be great. The question is as follows: Let $A ∈ M$ where $M$ is the set of $n \times n$ ...
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2 vector of equal length given how to find vector about which one vector reflected to that other?

Let $ w\in \mathbb R^n$ be vector of length $1$. $U$ is orthogonal space $w^\perp $ The reflection $r_w $ about $U$ is defined as follows if $v=cw+u$ , $u\in U$ then $r_w(v)=-cw+u$ Let $ u ,v$ be ...
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Inequality of vector norms with projections

Let $W$ be a vector subspace of $V$, a space with a dot product; $v\in V$. Let $p_W(v)$ be the orthogonal projection of $v$ onto $W$ and $w\in W, w\neq p_W(v)$. How can i prove that $||v-w|| > ||v-...
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why these two vectors are orthogonal

Let $E$ be a normed vector space. Moreover let $x \in E$ and suppose there are $y_1, y_2$ such that : $$\| x - y_1 \| = \|x - y_2 \|$$ Then my book say the following without any justifications (so ...
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isometry and orthogonality proof

If I have a relation (assuming $\vec{f}$ is one-to-one with $\det(\nabla \vec{f})>0$) appicable to all points from the domain of $\vec{f}$ which a regular region (a closed region with piecewise ...
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relation with range and null spaces and orthogonal complement [duplicate]

I'm trying to prove the following: $R(X)=\{\, Xa\ \mid a\in \mathbb R^{d+1} \,\}$ $\mathcal N(X^T)^\perp = \{\, a \in \mathbb R^{d+1} \mid a^T b = 0, b \in \mathcal N(X^T) \,\}$ $\mathbb R(X) = \...
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35 views

How to proof $\int_{-\infty}^{+\infty}\sin(w_1*t)\sin(w_2*t)\,dt = 0$ if $w_1 \neq w_2$

In my math script (signal theory) it says that two functions are orthogonal to each other when $\int_{-\infty}^{+\infty}s^\star(t)u(t)\,dt = 0$. Now I want to prove that $$\int_{-\infty}^{+\infty}\sin(...
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Orthogonality of the First Four Legendre Polynomials

Using the recurrence relation $$(n+1)P_{n+1}=x(2n+1)P_n(x)-nP_{n-1}(x) \ \ n\geq 1,$$ I've calculated the first four Legendre Polynomials as \begin{align} P_0(x)&=1 \\ P_1(x)&=x \\ P_2(x)&...
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1answer
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Problems with determining the orthonormal basis regarding an inner product $B$

Can somebody tell me where I made a mistake? How would you approach such an exercise? Was my way too complicated? $ B:\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R},\quad B(x,y)=\sum \limits_{i\neq j}^...
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1answer
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Evaluation of generalized Laguerre function integrals using orthogonality relations

(NB - I am not asking to be spoon-fed with complete solutions, just pointing out any useful transformations, or giving general pointers would suffice.) The orthogonality relation for generalized ...
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Bias-Variance OLS via eigendecomposition of projection matrix

I am struggeling to derive the squared-bias and variance based on an eigendecomposition for the OLS-procedure. The model Consider the univariate model $y_i = f(x_i) + \epsilon_i, \ i = 1, \dots, n$ ...
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1answer
34 views

Write the set of vectors that are orthogonal to $v$ as a linear combination of two unit vectors.

$$v = \langle 1, -\sqrt 8, -\sqrt 8\rangle \text{ is a vector.}$$ I know I have to find two unit vectors $u$ and $w$ so that any vector that is orthogonal to $v$ can be expressed as a linear ...
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1answer
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What is the intuition behind $\langle I- \frac{xx^T}{\|x\|_2^2},\frac{xx^T}{\|x\|_2^2}\rangle=0$?

For any non zero vector $x$ the following inner product is zero meaning that these two matrices are orthogonal to each other. $$\langle I- \frac{xx^T}{\|x\|_2^2},\frac{xx^T}{\|x\|_2^2}\rangle=0$$ ...
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1answer
33 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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Find the involutions in the indefinite orthogonal group O(2,1)

I would like to find the (linear) involutions that conserve the quadratic form $x^2+y^2-z^2$. Finding reasonable equations for the entries of the matrix is possible, but not particularly nice or ...
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Find a set of orthogonal vectors in $\mathbb{R}^{n}$ dimension with their components summing to zero

I am interested in finding a set of, say 3, orthogonal vectors whose components add up to zero. This is a concept that is used in statistics as well, but for the sake of this question, I wanted to ...
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43 views

Higher dimensional cross product equivalent

I'm working on a computer vision script for high dimensions that is highly reliant on the cross product in 3D, but as far as I know, it is only formally defined in 3D and 7D. However, experimentally, ...
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1answer
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determinant of two orthogonal matrices are zero when det(A)=det(B) [closed]

I am trying to solve following problem. Let $A,B \in \Bbb R^{n×n}$ be orthogonal matrices and $\det(𝐴)=\det(𝐵)$. . How can be proven that $𝐴+𝐵$ is singular? I know how to prove if the case ...
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$\{(X,Y)\in \Bbb R^3: |X|=|Y|=1, \langle X,Y \rangle=0\}$ is homeomorphic to $\text{SO}(3, \Bbb R)$

Let $R=\{(X,Y)\in \Bbb R^3: |X|=|Y|=1, \langle X,Y \rangle=0\}$ with $\langle\,\cdot\,,\,\cdot\,\rangle$ the Euclidean scalar product. Prove that $R$ is homeomorphic to $\text{SO}(3, \Bbb R)$. I ...
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1answer
111 views

Evaluating Fourier coefficients to complete a Laplace equation solution

While solving a PDE problem involving the Laplace equation in 3D, I arrive at the following summation relation when i substitute the only non-homogeneous boundary condition available $$ \sum_{m=1}^{\...
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2answers
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Show that $A$ is a difference between two orthogonal projections.

Let $V$ be a finitedimensional complex vector space. Linear operator $A \in L(V) $ is hermitian and unitary. Show that $A$ is a difference between two orthogonal projections. The questions seems ...
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Lift $O(\mathbb{Z}/p\mathbb{Z})$ to “something” in $O(\mathbb{Z}/p^2\mathbb{Z})$

So the question was from a result of Serre which basically says if $H$ is a closed subgroup of $Sp_{2n}(\mathbb{Z}_p)$ that maps surjectively onto $Sp_{2n}(\mathbb{Z}/p\mathbb{Z})$, then $H=Sp_{2n}(\...
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1answer
68 views

If $Q$ is a proper orthogonal transformation matrix, deduce that $\det(1-Q)=0$.

Show that if $Q$ is orthogonal transformation matrix, then $Q^t(Q-1)=(1-Q)^t$. Deduce that if $Q$ is also proper, then $\det(1-Q)=0$. Hence show that transformation has nonzero vector that has the ...
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1answer
52 views

If $(H_λ)_{λ≥0}$ is a spectral decomposition and $π_λ$ is the orthogonal projection onto $H_λ$, then $t↦π_λ$ is increasing and right-continuous

Let $H$ be a $\mathbb R$-Hilbert space. If $(\mathcal D(A),A_i)$ is a symmetric linear operator on $H$, write $A_1\le A_2$ if $$\langle A_1x,x\rangle_H\le\langle A_2x,x\rangle_H\;\;\;\text{for all }x\...
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1answer
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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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1answer
38 views

Is every orthogonal projection continuous?

Is there an example of a non-continuous linear operator $\pi$ on a $\mathbb R$-Hilbert space $H$ with $\pi^2=\pi$ and orthogonal null space and range? Clearly, if the range is closed, then $\pi$ is ...
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Proof of the orthogonality of a given matrix which involves binomial coefficients

I'm reading a paper (see pages 6 and 7 pf the pdf) which claims that a given $n+1$-dimensional square matrix $C$ is orthogonal. The $kj^{th}$ element of this matrix is given by: $$C_{kj}=2^{-\frac{n}{...
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2answers
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How to calculate norm of matrix using any orthogonal basis?

Show that for $X \in \mathrm{M}_n(\mathbb{C})$ and any orthonormal basis $\{u_1, \ldots , u_n\}$ of $\mathbb{C}^n$, we have $$\|X\|^2=\sum_{j,k}^n|\langle u_j,Xu_k\rangle |^2.$$ My Attempt: I ...
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1answer
24 views

Why can't the row space and nullspace be two lines in R3?

I'm learning Linear Algebra using Gilbert Strang's lectures and at lecture 14 he said the following: "Imagine two perpendicular lines in R3. Can they be the row space and the nullspace? No.". So the ...
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2answers
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Understanding a disprove example on Orthogonality subject

I was trying to disprove the following thoerem: If $A$ and $B$ are subspaces of $\mathbb{R}^n$ then $(A\cap B)^\bot = A^\bot \cup B^\bot$. I know that the theorem is not true, but my book gave the ...
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1answer
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Let $ w_1,w_2,\dots,w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$

Let $w_1, w_2,\dots, w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$ How do I find the scalars $a_1, a_2$, and $a_n$ ?
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Orthogonal projectors inequality

Let $V$ be a vector space and $V_1 \subset V_2$ two subspaces of $V$. Then, denoting by $P_1$ and $P_2$ the orthogonal projectors on $V_1$ and $V_2$ respectively, it holds for any matrix $A$ of ...
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24 views

Gram--Schmidt ortogonalization for complex functions

I am trying to perform the Gram--Schmidt orthogonalization on a set of continuos complex functions. Numerical solution would be quite sufficient for me. I have three continuous functions of complex ...
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2answers
47 views

Euclidean distance of set of orthogonal vectors

Let's define $x$ as a vector in $\mathbb R^n$ Let's define $V$ as the set of all vectors orthogonal to $x$, i.e $V$={$y$ in $\mathbb R^n$|$x·y=0$} Let's define $z$ as another vector in $\mathbb R^n$ ...
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Give the conditions on the $m\times1$ vector so that a matrix $H$ is orthogonal

Give the conditions on the $m\times 1$ vector $x$ such that the matrix $H=I_m-2xx'$ is orthogonal. The only solution I have found and verified is $x$ could be the zero vector; however, I know there ...
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1answer
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Finding the orthogonal complement of a subspace of $\mathbb{R}^4$

Let $V$ be the vector space $\{(x,y,z,w)\in\mathbb{R}^4:x+y-z=0\ \text{and}\ x+y+w=0\}$. Then, what would be a basis for the orthogonal complement of $V$. I think we have to find the null space of ...
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2answers
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Why are $(\,f_1,…,f_n)$ linearly independent if $\|\,f_k-e_k\|_2<\dfrac{1}{\sqrt n}$, where $(e_k)$ is an orthonormal basis? [duplicate]

Let $(e_1,...,e_n)$ be an orthonomal basis and $(\,f_1,...,f_n)$ vectors such that $\|f_k-e_k\|_2<\dfrac{1}{\sqrt n}\,\forall k\,$. I'd like to show that $(\,f_1,...,f_n)$ are linearly independent....
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For $n$ by $n$ matrix A, prove if dimension of row space > $n/2$, then $A^2 \ne 0$

I'm trying to prepare for an exam and came across the following question: Given $n \times n$ matrix A, let U represent row space and W represent column space. A) Prove: $W \subseteq$ $U^{\...
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Gram-Schimdt get the Legendre polynomial

When I apply the Gram-Schmidt algorithm I don't understand why I don't get the Legendre polynomials. When I apply this algorithm I always get monic polynomials whereas the Legendre polynomials aren't ...
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2answers
29 views

Proving $\dim(E_0) \geq n - k$

I found a question from an old exam which I am not really able to wrap my head around. The questions states: Given $k<n$ and $v_1, v_2, ..., v_k \in \mathbb{R} ^n$ non-zero vectors, orthogonal to ...
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2answers
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How do you find a point Q on the line L such that PQ is perpendicular to L

P is the point (1,1,1) and the line L is given by the equation x ¯ = t ( 1 ...
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1answer
42 views

calculate orthogonal matrix

Given matrix $A(m\times n)$ find matrix $B(n\times m)$ that fulfill the equation $A\,B=0\,(m\times m)$ mean orthogonal m less then n
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1answer
28 views

Linear Algebra, orthogonal columns and length

Suppose A is a 3x3 matrix whose columns are orthogonal and the length (two-norm) of each column equals 4. Then what is $A^T*A$? ...
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1answer
19 views

Question regarding orthogonality and linear independence

I am new to linear algebra and was a bit confused regarding the following… Any feedback would be really appreciated... True or false? If $K$ is a non-empty set of vectors in $R^n$, then $(K^\bot)^\...
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1answer
18 views

Question regarding the algebra of norms

I am new to linear algebra, as am having some doubts regarding the following question: True or False $u,v \in R^n$ $\left\lVert u\right\rVert=\left\lVert v\right\rVert$ if and only if u+v and u-v ...
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1answer
13 views

Two statements regarding orthogonal unit vectors and orthogonal complements respectively

I am new to linear algebra, and I was confused regarding the following question. I would really appreciate it, if anybody could give some feedback... True or False? $\left(\frac{1}{\sqrt14},\frac{-2}...
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2answers
24 views

Finding orthogonal basis in $\mathbb R^4$ from given vectors

I have two subsets of $\mathbb R^4$ $S=((-1,0,1,1),(0,1,1,1),(1,0,0,1))$ and $T=(x,y,z,x-y+2z)$ I've proved that T is a subspace of $\mathbb R^4$ and that S is a basis for T. So far, so good! I ...
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2answers
40 views

The Order of Orthogonality [closed]

I would like to show that $B\subset A$ implies $A^{\bot}\subset B^{\bot}$. Note the meaning behind this: The bigger a subset, the smaller its orthogonal should be. Let $x$ be in the complement of A. ...