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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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Decomposition of an unbounded operator

Suppose $H=H_1\oplus H_2\oplus...$ is an orthogonal decomposition of a Hilbert space $H$. Let $T:\mathrm{dom}(T)\subseteq H\to H$ a densely defined linear operator (it could be not bounded). For each $...
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Find which conditions must parameters $a$ and $b$ meet so there's exist an orthonormal basis

In $\mathbb{E^3}$ we have the plane $\pi:x-y+z-3=0$, the line $r:(2,0,1)+t(1,1,0),\ t\in\mathbb{R}$, and the point $P=(3,0,3)$. Which conditions must parameters $a$ and $b$ meet so there's exist an ...
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1answer
35 views

Orthogonal projection of the vector $p=(1,0,0,0)$ onto the subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$

I got a subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$ and I want to make an orthogonal projection of a vector $p=(1,0,0,0)$ onto $W$ and onto the orhhogonal complement of $W$. ...
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1answer
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(real symmetric matrices) does orthogonality of eigenvectors (distinct eigenvalues) depend on choice of basis?

For some reason I cannot wrap my head around this one. My schoolbook begins the chapter on eigendecomposition of real symmetric matrices by stating that eigenvectors from distinct eigenspaces are ...
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2answers
30 views

Projection and the Orthogonal Projection

Given $$ P = \left \{ (a,b,c,d) \in \mathbb R^4 \mid a + b + c + d = 0 \right \} $$ find $ P^\perp $. Am I right if I multiply $$ P^T P = 0$$ $P$ orthogonal is only the zero vector?
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Orthogonal Projection and Inner Product Space

Prove: Let V be an inner product space. $W⊂V$ and $v∈V$. Let $w∈W$ be an orthogonal projection. Then for every $u∈W ; ||w-v||<= ||u-v||$. I really do not have a clue on how to solve this.
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Is this property of a set of vectors is independent of the basis choice?

Let $\{v_1,\ldots,v_{n}\}$ be a set of orthonormal vectors in $\mathbb{R}^n$. If these vectors have this property that for some basis $\{a_1,\ldots,a_{n}\}$, for any $j\in \{1,\ldots,n\}$, $$\sum_{i=...
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1answer
20 views

Gradient descent orthogonal steps

For the steepest descent algorithm it's stated that Since $\alpha_k$ minimizes $\alpha\mapsto f(x_k + \alpha p_k)$ it follows $$ \nabla f(x_k + \alpha_k p_k)^Tp_k=0. $$ where $p_k = -\nabla f(x_k)$....
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Find basis of orthogonal complement of space W

I am supposed to find a basis of orthogonal complement of space W = $[(1,1,0,1),(0,1,0,1), (0,0,0,1)]$ . I have already found an orthogonal basis of W, which is $[(1,1,0,1), (-2/3,1/3,0,1/3) (0,-1/2,...
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Is there an elegant way to define orthogonality (and/or angles) without inner products, metrics, or norms?

I was wondering if there is an elegant and intrinsic way to define orthogonality on vectors without introducing inner products? Obviously "elegance" is subjective, so I'll try and give a sketch of the ...
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1answer
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Intersection and sum of closed sets and their orthogonal complements

I don't understand in the answer of 11b, why does that result follows from 11a? I know that the orthogonal complement is always a closed set, and that for closed sets, the orthogonal complement of ...
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orthogonal complement of tensor product

I am studying Functional analysis. And I don't understand an equation that is followed as below : If $S_1$ and $S_2$ are vector spaces, then $(S_1 \otimes S_2)^\perp = (S_1 ^\perp \otimes S_2) + (...
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summation and orthogonality assumption

The link has the problem that is supposed to be solved to the left and my solutions from earlier problems (1) and (3) to the right I am a bit confused as what do to with my earlier solutions? I did ...
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Proving equality for orthogonal matrices [duplicate]

Let $A$ be an orthogonal matrix, i.e. $A^T=A^{-1}$. Prove that $A(x\times{y})=\det(A)(Ax\times{Ay})$. I know that orthogonal matrices preserve distance, angles and orthogonality of vectors and I have ...
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getting a point belonging to a specific feasible region

Assume we have a bunch of linear inequalities. For example each of the inequalities look like this: $$2x_1+3x_2+x_3-5x_4<=10$$ So, the matrix form is: $$Ax<=b$$ in which A belongs to R^{m*n}, x ...
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1answer
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Basis for the subspace W orthogonal to a line

Let $V$ be the vector space of polynomials of degree $≤2$ with inner product given by: $\left \langle f,g \right \rangle=\int_{0}^{1}f(t)g(t)dt$ Let $f(t)=t+2$, $g(t)=t^{2}-2t-3$, and $h(t)=2t+1$. ...
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1answer
45 views

Finding the basis for vectors perpendicular to a plane

Let $W$ be the intersection of the two planes $\pi_{1}$ and $\pi_{2}$ defined by: \begin{align} \pi_{1} &=\{(x,y,z) \mid x−y−z=0\} \\ \pi_{2}&=\{(x,y,z)∣x+2y+z=0\}. \end{align} Find a basis ...
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1answer
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About the pseudoinverse $A^{+}$ in Gilbert Strang's “Linear Algebra and its Applications 2nd Edition”.

I am reading Gilbert Strang's "Linear Algebra and its Applications 2nd Edition". He wrote "All solutions of $A \overline{x} = p$ share this same component $\overline{x_r}$ in the row space, and ...
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2answers
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About orthogonal complement in Gilbert Strang's “Linear Algebra and its Applications 2nd Edition”

I am reading Gilbert Strang's "Linear Algebra and its Applications 2nd Edition". On p.137(3.4 The PSEUDOINVERSE AND THE SINGULAR VALUE DECOMPOSITION), he wrote "any vector can be split into two ...
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An Orthogonality Problem of Eigenfunctions of homogeneous Fredholm equation

Suppose we have a integral equation $$\int_{-1}^1 \frac{\text{sin }c(x-y)}{\pi (x-y)}\psi(y)dy=\lambda \psi(x),\quad|x|\le1.$$ By the Fredholm equation theory, we know that this equation has ...
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1answer
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Orthogonal bases of the vector space $\mathbb{Z}_2^4$

Let $\mathbb{Z}_2$ be the two element field $\mathbb{Z}/2\mathbb{Z}$. The vectors $e_0 = \langle1,1,1,1\rangle$, $e_1=\langle1,1,0,0\rangle$, $e_2 = \langle1,0,0,1\rangle$, $e_3 = \langle1,0,1,0\...
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2answers
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Integrals of products of sines and cosines with arbitrary periods

I am currently studying the Fourier series, which involves integrals of products of sine and cosine functions. Because sine and cosine are orthogonal, we have been using the following facts to help us ...
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How to reproduce this influence statistic found in this Taguchi method example?

Objective: to recreate the influence values given in this online tutorial. The context for this question is from [Design of Experiments][1] from Taguchi. Thank ...
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1answer
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How do I find a unit vector orthogonal to a line?

I am looking at an academic paper. In one section of it I need to find a line orthogonal to the between two points. The paper says: Given a point p ∈ δΩ the normal direction is computed as ...
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1answer
25 views

Decomposition of self adjoint elements by positive elements

Let $a \in A$ be a self adjoint element of a $C^*$ algebra. There exists positive elements $a_+, a_-$, such that $$a=a_+ - a_{-} $$ $$a_+a_-=a_-a_+=0$$ Is the statement true? This is ...
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Understanding the orthogonal projection vector derivation

As you can see below, $z$ is the projection of $x$ onto $y$... I am trying to derive the orthogonal projection formula based on things I already know. Calculating $cos(\theta)$ is trivial... $$cos(\...
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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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1answer
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Relation between rank, nullity and orthogonal complement

Question Let $A\in\mathbb{R}_{d\times d}$ be some square, non-invertible matrix. Prove that if $b\perp\ker(A^T)$, then the non-homogenous system $Ax=b$ has $\infty$ solutions. Background I got ...
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1answer
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How do we show two functions have orthogonal actions on a set or group?

If we take initially $X=\{x\in \Bbb Z[\frac12]\setminus0\}$ then we can see pretty quickly that from any given starting number, the actions of the two functions on $X$ will generate the whole set: $f(...
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1answer
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Orthogonal vector to a plane using a different inner product

Considering $\mathbb{R}$ with the inner product $$\langle(a_1,a_2,a_3),(b_1,b_2,b_3)\rangle=2(a_1b_1+a_2b_2+a_3b_3)-(a_1b_2+a_2b_1+a_2b_3+a_3b_2)$$ Then, how could we find the set of vectors ...
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3answers
43 views

If $\|u+tv\| \ge \|u\|$ for all $t$, prove that $u \cdot v=0$

Let $u, v \in \mathbb R^n$. Prove that if $$\|u+tv\| \ge \|u\|$$ for all $t \in \mathbb R$, then $u\cdot v=0$ (vectors $u$ and $v$ are perpendicular). I tried writing $v$ as $(n+xu)$, where $u\cdot n=...
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Proof of Gram-Schmidt process using strong induction

The Gram-Schmidt orthogonalization of a linearly independent set $S=\lbrace v_1,v_2,\dots,v_p \rbrace$-- assuming finite-ness for convenience-- is given by $u_1=v_1$ and $k>1\implies u_k=v_k-\sum_{...
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1answer
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Is relationship of orthonormality & orthogonality equivalent to that of squares & rectangles?

To confirm my analogy, I am asking if I can consider every orthonormal basis to be an orthogonal one (but not vv) in the same sense that every square is a rectangle (but not vv). I believe the answer ...
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1answer
36 views

Finding orthonormal basis of subspace spanned by two functions

From S.L Linear Algebra: Let $V$ be the subspace of functions generated by the two functions $f$, $g$ such that $f(t)=t$ and $g(t)=t^2$. Find an orthonormal basis for $V$. In this case, $V$ is ...
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1answer
17 views

Orthogonal basis of a quadratic module

I'm studying quadratic forms and I have to prove that every quadratic module $(V,Q)$ has an orthogonal basis, where $V$ is a vectorial space on $\mathbb{K}$ and $Q$ a quadratic form on $V$. I'm ...
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1answer
11 views

Length of vector function achieving local minimum at $t=t_0$

If a differentiable vector-valued function $r(t)=<x(t),y(t),z(t)>$ has a length $|r(t)|$ that achieves a local minimum at $t=t_0$, then $r'(t_0)\cdot r(t_0)=0$. This statement is apparently ...
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1answer
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Calculate $\inf _{a, b, c \in \mathbb{R}} \int_{-1}^{1}\left|x^{3}-a x^{2}-b x-c\right|^{2} d x$

Calculate $$\inf\limits_{a, b, c \in \mathbb{R}} \,\int_{-1}^{1}\left|x^{3}-a x^{2}-b x-c\right|^{2} \mathrm d x$$ I am new to Hilbert space, I see similar questions used the formula: $\langle f, g\...
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1answer
35 views

Orthogonal Property of Legendre Polynomials

How can I get $$nu_{n} + (n-1)u_{n-1},$$ where $$u_{n} = \int_{-1}^{1} x^{-1}P_{n-1}(x) P_{n}(x)\, \mathrm {d}x\; ? $$ I did many search, and also I did try by myself. But without a success. Is ...
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Prove that the optimal solution of a fitting term does not effect by the outlier

I'm having difficulty in proving the solution of this problem: The given vectors $\mathbf{a}_1$, $\mathbf{a}_2 \in \mathbb{R}^M$ and the variables $\mathbf{b}_{1},\mathbf{b}_2\in\mathbb{R}^M$. ...
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Theorem of contiguous bases for quadratic forms

I'm studying theory of quadratic forms on Serre's book "A course in Arithmetic". I'm trying to prove Theorem 2, chapter 4, which says that given two orthogonal bases $E$ and $F$ for a quadratic module ...
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1answer
35 views

Struggling to apply algebra to an inner product

I'm stuck in the following problem. I feel there may be an elegant solution that's avoiding me, as it's obvious choosing $a$ such that $x-ay$ orthogonal to y will minimize $x-ay$, but I'm getting ...
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1answer
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Scalar Products and Projections

Proposition: Let w $\in V$ so that V is a vector space over $\mathbb{R}$ and $||w|| \neq 0$. For every v in V there exists a unique $c\in \mathbb{R}$ so that $v-cw$ is perpendicular to w. My Proof: ...
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Proof of the Normality of Bessel

This is a my proof of the normality of bessel functions . But I want to check it and I prefer another proof if exist. Help me please. The Bessel function $J_{\nu}(x)$ satisfies the following ...
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1answer
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how can vectors not be of unit norm

I have a Linear homework questions asking what the QR factorization of a matrix A whose columns are orthogonal but not of unit norm might look like. I reread the section in textbook about norms, but ...
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1answer
51 views

A “unique” solution to an equation over the orthogonal matrices?

Set $D=\text{diag}(-1,1,1,\dots ,1)$ be an $n \times n$ real diagonal matrix (where $D_{11}=-1$ and $D_{ii}=1$ for $i>1$). Let $R,Q$ be special orthogonal matrices, satisfying $RDQ=D$. Is it ...
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2answers
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How is $(x_1,x_2)$ normal to $x_1w_1 + x_2w_2 = y$?

Note: this question is related to the maths of Neural Nets, if you need clarification about the question do comment. Raul Rojas' Neural Networks A Systematic Introduction, section 8.1.2 relates off-...
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1answer
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The orthogonal complement of the orthogonal complement from “Linear Algebra Done Right”

The following content is from "Linear Algebra Done Right" by Sheldon Axler Corollary: Suppose $U$ is a finite-dimensional subspace of $V$. Then $$U = (U ^\perp)^\perp.$$ We need to prove the ...
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28 views

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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1answer
36 views

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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2answers
31 views

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 $n \times n$ ...