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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Is there any relation between these polynomials and a set of orthogonal polynomials?

as you know for every set of orthogonal polynomials as $P_0(x)=1, P_1(x), P_2(x),..., P_n(x), ...$ we have exactly $n$ real roots for $P_n(x)$ and also the fact that $\int_a^b P_i(x)P_j(x)d\alpha(x)=0$...
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Is it true that $T$ is orthogonal if and only if $T$ is isomorphism?

I want to prove the following: Let $V$ be a finite dimensional vector space with inner product, then $T$ is orthogonal if and only $T$ is an isomorphism I think the sufficiency could be true because ...
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Why does $T(x)=2x$ keep angles but is not an isometry? [closed]

I would like to know why a defined operator $T$ on $V$, as following: $T(\alpha)=2\alpha$ preserves angles but is not an isometry?
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How to calculate angle in non-orthogonal system (B angle at 45 degree, C angle attached on B) based on orthogonal angles

I am trying to find a formula to recalculate rotation from orthogonal BC system to defined non-orthogonal BC system In which B is shifted by 45 degree. Graphic example below: non-orthogonal system B ...
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Orthogonal tranformation over invariant subspace

let $V$ be a euclidean space. $T:V\to V$ be an orthogonal linear transformation. $W\subset V$ is a $T$-invariant subspace. I need to prove 2 things: A. $T\bigl|_W$ is orthogonal so I said that if ...
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28 views

Finding orthogonal projection on Hilbert space

Let $H = L^2(−1, 1)$ and $L \subset H$ be the set of all continuous functions such that $f(0) = 0$. Find the orthogonal projection $P : H → \bar{L}$. My thoughts on this: We say $g=P_{\bar{L}}(f)$ ...
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Help me prove that an orthogonal operator preserves length [closed]

Help me prove this please. Let $V$ be a finite-dimensional vector space with inner product $\langle \cdot, \cdot \rangle.$ Let $T$ be a linear operator on $V.$ If $T$ is orthogonal or unitary, then $...
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Are perpendicular vectors always in different subspaces?

So I understand when two subspaces are considered perpendicular and what it means for vectors to be perpendicular/orthogonal. The question I have is, if two vectors are perpendicular, do they always ...
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How would I go about solving this projection matrices question?

The linear algebra course that I'm taking has provided us with examples of how to find the orthogonal projection of a vector onto a subspace given the orthogonal basis of that subspace, however, I am ...
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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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If $V$ is right-orthogonal, does it hold $\langle AV,BV\rangle_F=\langle A,B\rangle_F$?

Let $A,B\in\mathbb R^{m\times n}$. It's easy to see that for the Frobenius inner product it holds $$\langle A,B\rangle_F=\operatorname{tr}B^\ast A=\operatorname{tr}A^\ast B.\tag1$$ So, if $U\in\mathbb ...
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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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20 views

Rank of a left-orthogonal decomposition $A^T=UB^T$

I've got a rather simple question, but couldn't find an answer to it: Say $A\in\mathbb R^{m\times n}$ can be decomposed according to $$A^T=UB^T\in\mathbb R^{n\times m}\tag1$$ for some left-orthogonal (...
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Using a Householder reflector to determine column of a matrix

In my numerical analysis class, we were given a set of optional problems to prepare for the final. I'm having trouble understanding one of the questions: Given a Householder reflector $H$ such that $...
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42 views

Best approximation of a vector $x=\begin{bmatrix}2&2&0&0\end{bmatrix}^\tau$ by the vectors in $M^\perp$, where $b\in M$ is given

In a unitary space $\Bbb R^4$, subspace $M=\operatorname{span}\left\{b=\begin{bmatrix}1\\1\\1\\1\end{bmatrix}\right\}\leqslant\Bbb R^4$ is given. Find the best approximation of the vector $x=\begin{...
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Does $A^TA=I$ imply $AA^T=I$?

Let $m,n\in\mathbb R^{m\times n}$ with $$A^TA=I_n\tag1.$$ I wonder whether this implies that $$AA^T=I_m\tag2$$ or if we can show it at least in the case $m=n$. EDIT: I was hoping for a proof which ...
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Form a new base with two vectors

I need to make a new orthonormal base in $\mathbb{R^3}$ given $(2,7,5)$ and $(4,1,3)$ so that it makes $(\widehat{e_1}, \widehat{e_2}, \widehat{e_3} )$. But $ \widehat{e_1} $ has the same direction of ...
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Prove (a ∧ b) · c does NOT equal 0 if and only if a, b and c are linearly independent

Prove that $(a ∧ b)\cdot c\ne0\iff a, b$ and $c$ are linearly independent, where $a,b,c\in\Bbb R^3$. In the first part of the question, I proved that $(a ∧ b)$ is orthogonal to $a$ and to $b$, ...
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Show that each vector in an n-dimensional vector space can be represented as the summation of its components along the orthonormal basis.

Show that in an n-dimensional vector space V over the universal set with orthogonal basis {$a_1, a_2,..., a_n$}, each vector B can be expressed as: B = $\frac{<B,a_1>a_1}{||a_1||^2}$ + $\frac{...
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Norms and triangular inequality in Hilbert spaces

My question is about an intuition and it arises from the following problem. Let $C$ be a convex set (nonempty and closed) in $R^d$, and $P_C$ be the orthogonal projection onto $C$. We need to show ...
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If $Z_n = X_n + Y_n$ for $X_n\in M$ and $Y_n\in N$ then $(X_n)$ and $(Y_n)$ converge

Let $H$ be a Hilbert space (infinite dim) with $M,N\subset H$ being closed subspaces satisfying $N\subset M^\perp$. I'm trying to show that $M+N$ is closed. If $(Z_n)_{n=1}^\infty \subset M+N$ is a ...
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Check if the system: $\sin \left(\frac{2\pi n x}{b-a} \right), \cos \left(\frac{2\pi n x}{b-a} \right)$is orthogonal over the interval [a=12, b=14].

Check if the system: $\sin \left(\frac{2\pi n x}{b-a} \right), \cos \left(\frac{2\pi n x}{b-a} \right)$is orthogonal over the interval [a=12, b=14]. My work thus far: $$\begin{align} \int^{14}_{12} \...
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if the column set of A is orthogonal, and C is diagonal such that C = B^T*A, is the column set of B Orthogonal?

So, I have to check whether or not the column set of a matrix B is orthogonal, given the fact that the column set of the matrix A is orthogonal, and the fact that the diagonal Matrix C = B^T*A. my ...
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Different methods to find orthogonal complement

Say I have a set of vectors and I want to find a basis for the orthogonal complement. I know a way to do this is to just calculate the dot product of each given vector with a general vector. This way ...
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Proving the orthogonality of $\sin\frac{2\pi x}{\pi-e}$ and $\cos\frac{2\pi x}{\pi-e}$

I want to prove the orthogonality of the functions: $\sin\left(\dfrac{2\pi x}{b-a}\right)$ and $\cos\left(\dfrac{2\pi x}{b-a}\right)$, where $b=\pi$ and $a = e$ My work: $$\begin{align} \int^{\pi}_{...
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36 views

Orthogonal to Orthonormal basis [closed]

I am having trouble completing the last step of my problem below: Let $V= \beta [0,2] = \{ $All continuous function $f:[0,2] \rightarrow \mathbb{R} \}$. Let $W = span(1,x,x^2) \subset V$. Find ...
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Explain why $4$ cannot be replaced by $5$ in part a)

a) Construct a Latin square of order $8$ in which the submatrix formed from the first $4$ rows and $4$ columns is the addition table for $Z_4$. b) Explain why $4$ cannot be replaced by $5$ in part a) ...
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If subspace $M$ is invariant under operator $A^+$ on $V^+$, prove that $M^\perp \cap V$ is invariant under operator $A$ on $V$.

Exercise 77.6, page 153 from PR Halmos's Finite-Dimensional Vector Spaces, 2nd Ed: Given that $A$ is an operator on a real vector space $V$, and that a subspace $M$ of the complexification $V^+$ is ...
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28 views

dim(nul(A)) = dim(nul(A^T))?

I am currently trying to prove that the union of an orthogonal subspace $W$ and its orthogonal complement $W^\perp$ span $\Bbb R^n$. In order to do this, I am trying to use the Rank-Nullity theorem. ...
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31 views

Diagonalisability of matrices

There are $2 \times 2$ matrices $A$, $B$, $C$, and $D$ with following properties. $A$ is symmetric and negative definite. $B$ is orthogonal and $\det(B)=1$. $C$ is orthogonal and $\det(C)=-1$. $D$ is ...
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How to show search direction is $A$-orthogonal to the previous ones in conjugate gradient method?

Consider conjugate gradient method as follows for solving $Ax=b$ where $A$ is a positive definite matrix: $$ x_0 = 0 , r_0 = b, p_0 = r_0\\ \textbf{for} \quad n=1, 2, \cdots \quad \textbf{do}\\ q_n = ...
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missing basis and finding coordinates of vector with respect to basis

Vectors $w_1, w_2, w_3$ form an orthogonal basis for $R^3$. Given that $w_1 = \begin{pmatrix} 2\\3\\5 \end{pmatrix}$, what are the coordinates of the vector $v=\begin{pmatrix}0\\1\\2\end{pmatrix}$ ...
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If two vectors intersects at null space only, does that mean they are orthogonal? [closed]

By intersecting I mean that they share only points that lies on their null space. I probably thinking only about a case when their null spaces are zero vector. But I couldn't come up with an example ...
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Question about Orthogonal Complements

I had difficulties understanding this question. Could you give me some advice how to approach this question? I couldn't create the relationship between the given features. Let $x =\begin{bmatrix}1&...
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Exercise on orthogonality in a Hilbert space

Problem Let $V$ and $W$ two subspaces of a Hilbert space $H$, with $dim(V)= m -1$ and $dim(W) = m$, with $m \geq 1$. Prove that if $V=span\left\{e_1, ..., e_{m-1}\right\}$, then there exists a vector ...
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36 views

How to Divide One Vector by Another

I have the following question and solution and I don't understand how to get from $$ \frac{[1,1,−1]⋅[−1,−2,−1]}{[−1,−2,−1]⋅[−1,−2,−1]} $$ to $$-\frac{2}{6}$$ Any ideas much appreciated. Thanks! Let $...
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Is $f - 3I$ an isomorphism if $f$ is orthogonal?

Decide if the following statement is true or false by briefly justify the answer. Let $(V, \phi)$ be a real Euclidean space of dimension $n$, and let $f: V \to V$ be an orthogonal operator. ...
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Expansion and orthogonality

It is stated various places that as the Legendre functions $$ \sum_{m=0}^\infty P_m(\cos\theta) $$ are orthogonal, this enables us to expand any function of $\theta$ into a set of these functions. ...
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Are the eigenvectors of a normal matrix are orthogonal regardless of the eigenvalues?

*I know the sentence that for normal matrices, the eigenvectors, of different eigenvalues, are orthogonal. Yet, I also know that every normal matrix is unitary diagonalaizable. Also, the columns ...
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Determine if 2 eigenvectors are orthogonal.

Let $$ T:V \to V $$ Be a linear operator. And: $$ 0 \neq u,v \in V $$ Such that: $$ T(v) = \lambda_1v, \quad T(u) = \lambda_2u $$ NOTE: It may be that $\lambda_1 = \lambda_2$ The question ...
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Operator norm of $T\colon H\to H$ if we know $\|T|_{W}\|$ and $\|T|_{W^{\perp}}\|$, where $W\subset H$ is a closed subspace.

Suppose that $H$ is a Hilbert space and $W\subset H$ a closed subspace. Let $T\colon H\to H$ be an operator. Suppose that $T_{W}\colon W\to H$ and $T_{W^{\perp}}\colon W^{\perp}\to H$ are bounded. ...
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Induction on $g(T)=\sum_{i=1}^k g(\lambda_i)T_i$.

Let T be a normal operator on a finite-dimensional complex inner product space V. If g is a polynomial, then $g(T)=\sum_{i=1}^k g(\lambda_i)T_i$. I know that $T_i T_j =0$ for $ i \not= j$ $(T_i$ is ...
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Orthogonal projection find a vector such that $\|T(v)\|>\|v\|$

Let T be a linear operator on a finite-dimensional inner product space V. If T is a projection such that $\|T(x)\| \leq \|x\|$ for all $x \in V$. Prove that T is an orthogonal projection. I want to ...
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Spectral norm of projected matrix

Let $M_{n,m}$ be the set of real matrices of $n\times m$, and let $T:M_{n,m}\to M_{n,m}$ be a orthogonal projection operator, i.e., $T$ is such that for any $A,B\in M_{n,m}$ $$T(A+B)=T(A)+T(B),$$ $$T(...
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Adjoint of right shift operator on orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\ell^{2}(\mathbb{N})$

I'm sorry if this question is a duplicate. Suppose $(e_{n})_{n\in\mathbb{N}}$ is the usual orthonormal basis of $\ell^{2}(\mathbb{N})$. We can define an operator $v\colon H\to H$ by $ve_{n}:=e_{n+1}$....
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1answer
25 views

Isometry from $\mathbb R^m$ to a subspace of $\mathbb R^n$

Let $\{w_1,\dots,w_m\}$ be a basis of a subspace $U$ of $\mathbb R^n$ and $\{v_1,\dots,v_m\}$ be an orthogonal basis of $\mathbb R^m$. Is there a canonical way to obtain a linear transformation $T:\...
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35 views

How to prove polarisation identity?

Given that $A$ is a complex $n\times n$ Matrix. How do I prove that $\langle A(u + v) , u + v\rangle − \langle A(u − v), u − v \rangle = 2\langle Au, v \rangle + 2\langle Av, u \rangle$ ? I am stuck ...
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Rayleigh quotient and orthogonality.

I'm studying chapter 6 of David Lay's Linear algebra and Its Applications and I'm having a little trouble understanding what this exercise has to do with orthogonality. The following has been ...
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66 views

Multipole expansion of solution to the Poisson equation

In electrodynamics I have seen the following: Let $\phi$ be a solution to the Poisson equation $-\Delta \phi= \rho$, and assume that $\rho$ is compactly supported. Then we can expand $\phi$ as the ...
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Real and imaginary parts of basis of complex eigenvectors to form base of real eigenvectors?

Given a basis of complex eigenvectors for, say, a $2 \times 2$ symmetric matrix $A$ (which hence has real eigenvalues). Can one generate a basis of real(-valued) eigenvectors from the real and ...

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