Questions tagged [orthonormal]
For questions related to orthonormality. A set of vectors in an inner product space is called orthonormal if each vector is a unit vector, and vectors are pairwise orthogonal.
1,255
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Orthonormal Basis of $L^2[0,1]$ having indicator functions?
I was asked to construct an ONB of $L^2[0,1]$ having functions taking at most two values. By suitably scaling, I can think them as indicator functions. So question boils down to finding countably many ...
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Orthogonal Bases and Fourier Approximation
I am confused about how my professor resolved a fourier approximation. The question was:
Find the fourth-order fourier approximation of: $$f(x) = \sin (5x)$$
on the interval: $$0\le x \le 2\pi$$
He ...
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27
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The converse for the basis of tensor product of Hilbert spaces
I have been studying the tensor product of Hilbert spaces, and I know that if ${\psi_k}$ and ${\phi_l}$ are orthonormal basis of $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively, then ${\psi_k\otimes\...
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Absolute value of dot product of high-dimensional vector
My professor during the class mentioned (page 35) that for two vectors in high-dimensional space (say dimension $d$), we typically have
$$
\left|\boldsymbol{u}^T \boldsymbol{v}\right| \approx \frac{\|\...
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Riesz Representation Theorem & Weak closure of orthonormal basis in Hilbert space [closed]
So here it was shown in an infinite dimensional seperable Hilbert space $H$, with the orthonormal basis $E=\{e_1,e_2,\dots \}$ that $0\in \overline{E}^w$, where the right hand side denotes the weak ...
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Prove that which is an orthonomal basis of $L^2(R)$
Prove $$\left \{\frac{1}{π^{\frac{1}{2}}}\left (\frac{i-x}{i＋x} \right )^n\frac{1}{i＋x} \right \}_{n＝-\infty }^{n＝\infty}$$is an orthonomal basis of $L^2(R)$
I tried something like taking the inner ...
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Representation of closed linear span of an orthonormal set in Hilbert space
This is a follow-up of this post.
Here is the statement we want to prove.
Let $\{x_j\}$ be an arbitrarily indexed orthonormal set in a Hilbert space(possibly non-separable). Show that the closed ...
- 251
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Show sum of coordinates squared less than or equal to norm squared
I'm trying to show that if $S = \{v_1, \dots, v_q\}$ is orthonormal, then for every $v$ in a vector space $V$ we have
$$
\sum_{k=1}^q|\alpha_k|^2 \leq \Vert v \Vert_V^2
$$
where $\alpha_k = \...
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Finding an orthonormal basis with respect to an inner product.
Let
$A=\left(\begin{array}{lll} 2 & 0 & 1 \\\ 0 & 2 & 0 \\\ 1 & 0 & 2 \end{array}\right) \in \mathrm{M}_{3}(\mathbb{R})$
(a) Using the matrix $A$, we define $\langle x, x\...
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Functional Analysis trying to show that hilbert space equals $\sum \mathbb{F} e_j$
Let $H$ be a Hilbert space and $\{e_j: j \in I\}$ be a orthonormal basis i.e. $(e_i|e_j) = 0$ whenever $i \neq j$, $||e_j|| = 1$, and the closure of $\text{span}\{e_j:j \in I\}$ is equal to $H$. I ...
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Confusion regarding matrix representation properties
I am studying about matrix representation of finite groups. If the group is defined as
\begin{equation}
G=\{e,a,b,c,.....\}
\end{equation}
then the matrix representation is defined by the collection ...
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How to construct Legendre polynomials for $x_1,...,x_k$?
I'm trying to run a nonparametric regression to estimate the unknown conditional mean $E(Y|X_1=x^*_1,X_2=x^*_2)$ using data set $\{Y_i,X_{1i},X_{2i}\}_{i=1}^n$. This could be done by nonparametric ...
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Estimating mean curvature of a surface by two perpendicular curves of the surface
Let S be an oriented smooth surface containing a circle of radius 1 and a straight line, which intersect perpendicularly at a point $p\in S$. Show that if the Gauss curvature K of
S satisfies
K(p)=0, ...
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Why should eigenvectors be orthonormal if we start in an orthonormal basis?
As part of the development of degenerate perturbation theory, I think the author uses the following fact tacitly. I'll assume we're in a finite-dimensional space for simplicity. Suppose I have an ...
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How to define coordinates for a non-orthonormal basis
I have two non-orthonormal basis vectors, and I want to represent a third vector as a pair of coordinates using the aformentioned basis vectors. How would I do that? The dot product, which usually ...
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Trace of "quadratic form"-like expression involving orthonormal vector fields and their Jacobian
Let $f\in C^{\infty}(\mathbb{R}^D, \mathbb{R}^K)$ be a smooth function whose Jacobian matrix $Df$ has full rank on $M=\{x\in \mathbb{R}^D: f(x)=\vec{0}\}$. We take the convention that the rows of the ...
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How to block-diagonalize a skew symmetric matrix
I have encountered the fact on wikipedia that every skew-symmetric matrix can be block-diagonalized, where the matrix is in the form indicated by the following picture. I was wondering if there is a ...
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Does every infinite-dimensional Banach space admit an "almost orthonormal" infinite set?
Every infinite-dimensional Hilbert space admits an orthonormal basis. The basis has the three properties:
The elements are linearly independent
All elements have length $1$.
The pairs between any ...
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Uniqueness of reproducing property in Christoffel-Darboux Kernel polynomial
I am a beginner in studying the Christoffel-Darboux kernel polynomials that were defined as follows:
$$
K_n(y,x)=\sum_{k=0}^n p_k(y)p_k(x),
$$
where $p_0,...,p_n$ are orthonormal polynomials for a ...
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Geometric intuition of Euler-Maruyama scheme for Brownian motion on manifold
Let $M$ be a manifold defined by an implicit equation $f(x)=0$ for some smooth function $f:\mathbb{R}^D\to \mathbb{R}^K$, where $K=D-d$. Assume the Jacobian matrix $Df(x)$ has rank $K$ on $M$. ...
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Finding the minimum approximation with orthonormal basis of a given integral
With two continuous functions, the inner product is defined:
$$
\langle{f,g}\rangle = \int_0^1f(x)g(x)dx
$$
Here, I would like to find $h(x)$ such that the following integral can be approximately ...
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Example of a semi-orthogonal matrix with a condition on the norm of its rows
Let $m>n$. I would like to find an example of a matrix $M \in \mathbb{R}^{m \times n}$ such that the columns of $M$ are orthonormal vectors (for the euclidean norm), and such that the maximal ...
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orthogonal vs orthonormal matrices - what are simplest possible definitions and examples of each ??
I'm trying to understand orthogonal and orthonormal matrices and I'm very confused. Unfortunately most sources I've found have unclear definitions, and many have conflicting definitions!
Some sites ...
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Prove that, if $v\in F^\perp$ and $v\in F$ then $v=0$ ($F$ is the span of a finite orthonormal system)
Let $\left\{e_1, \ldots, e_n\right\}$ be a finite orthonormal system in a inner product space $(E, \langle\cdot, \cdot\rangle)$ and let $F=\operatorname {span}\left\{e_1, \ldots, e_n\right\}$. Prove ...
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Reproducing kernel Hilbert space induced by $k(x, y) = \delta_{x, y}$, where $\delta$ is the Kronecker delta
I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel
$$
k \colon X \times X \to \{ 0, 1 \}, \qquad
(x, y) \mapsto %\...
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Matrix representation of adjoint operators
I am studying linear algebra.
In the textbook I use, the book says that(Friedberg, 5th ed, p.356 theorem 10)
If $V$ is an f.d.i.p.s and $B$ is an O.N.B for $V$, $[T^{*}]_{B}=[T]_{B}^{*}$ holds for ...
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Write all the unitary vectors of $\mathbb{C}^2$ in terms of real parameters
Write all the unitary vectors of $\mathbb{C}^2$ in terms of real parameters. After so, write all the orthonormal basis of $\mathbb{C}^2$
I thought that for the first part, a vector $\vec{u}=(x_1,x_2)$...
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An inner product space that every closed subspace is complemented, is it must be separable?
In a comment of this answer someone said that
if an inner product space satisfy that every closed subspace is complemented then it has an (countable) orthonormal basis
and this answer claim that
H has ...
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Show that $A \in \mbox{SO} (3)$ and find an orthonormal basis, so that $A$ have canonical form
Let $$A = \frac{1}{15} \left( \begin{array}{rrr} 10 & 5 & 10 \\ 5 & -14 & 2 \\ 10 & 2 & -11 \end{array}\right)$$ Show that, $A \in \mbox{SO} (3)$ and find an orthonormal basis, ...
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Inner product with two different bases
Let $V$ be a finite-dimensional inner product space. If $B= \{ b_{1}, b_{2},\cdots, b_{n}\}$ is a basis for $V$, show that $B'=\{f_{1},f_{2},\cdots,f_{n}\}$ is also a basis for $V$ with property $\...
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Existence of a $q$-orthonormal basis of $\mathbb{R}^{r+s}$
For some propositions and proofs it was assumed that there exist a $q$-orthogonal (respect. $q$-orthonormal) basis of $\mathbb{R}^{r+s}\subset Cl_{r,s}$. Here $q$ is a quadratic form. Since we ...
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Searching for weighted-$L^1$ summable orthonormal basis of $L^2(0,\infty)$
so I was working on something and bumped into the following question:
Given some $a>0$, does there exist a complete orthonormal system $ (f_n)_{n \in \mathbb{N}} $ of $L^2(0,\infty)$ such that $\...
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Showing that $\{\varphi_{m,n} \}_{m \geq 1, n \geq 1}$ is an orthonormal basis for $L^2((a,b) \times (a,b)).$
Let $(a,b) \subseteq \mathbb R$ and $\{\varphi_n \}_{n \geq 1}$ be an orthonormal basis for $L^2((a,b)).$ Define $\varphi_{m,n} : (a,b) \times (a,b) \longrightarrow \mathbb C$ by $$\varphi_{m,n} (s,t) ...
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Let $T\colon V\to V$ over the IPS $V$ and $B$ be an orthogonal basis for $V$. Find the simplest connexion between $[T^*]_B$ and $([T]_B)^*$
Let $T\colon V\to V$ over an Inner Product Space $V$.
Let $B$ be an orthogonal basis for $V$.
Find the simplest connexion between
$[T^*]_B$ and $([T]_B)^*$.
So I know that if $B$ was an orthonormal ...
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A suspicion on orthonormal basis coefficients
Let's assume I have a $2$ dimensional vector space with inner product, and a basis where the inner product can be represented as
$$
\begin{bmatrix}x &y\end{bmatrix}
\begin{bmatrix}E &F\\F &...
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Determination of tangential and binormal vectors
I am struggling to find binormal and tangential vector, which are both perpendicular to normal vector, which I have found.
So I have a right-hand system ($\mathbf{e_x,e_y,e_z}$), there is a sketch ...
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span of maximal orthonormal sets in a Hilbert space
I am working through Walter Rudin's Real and Complex Analysis and I'm having trouble understanding something about Theorem 4.18:
I understand the proof of the theorem, but I'm confused about the ...
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Orthonormal matrix with the least column variance
How can I construct an orthonormal matrix in which the variance over the columns is minimal?
To be more formal, I would like to find a matrix $\mathbf{A} = [\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{...
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Using Gram-Schmidt to extend an orthonormal basis for a subspace
So, given a basis for $V$, I know how to use the Gram-Schmidt process to get an orthonormal basis.
My question is, suppose we have an orthonormal basis for a subspace $U$ of $V$. How can we extend ...
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Proof of Bessel inequality by orthogonal projectors
Prove that for every orthogonal family of projectors $(P_n)$ on Hilbert space $H$ $$\sum\limits_{n=1}^{\infty} ||P_nf||^2 \leq ||f||^2, \forall f \in H.$$
From it, prove Bessel inequality $$\sum\...
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An example of dense subspace of $\ell_2$
Below is my explicit example of a uniformly convex normed vector space which is not reflexive. Could you confirm if my understanding is correct?
Let
$$
\ell_2 := \big \{ (v_n)_{n \ge 1} \subset \...
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Example of an inner product space with no orthonormal basis
Let $X$ be an infinite-dimensional vector space with an inner product $(\cdot, \cdot)$. A system of non-zero vectors $B = \{ x_\alpha \}$ from $X$ is called orthonormal if
$$
(x_\alpha, x_\beta) =
\...
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answer
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Prove the squares of a particular coordinates of all vectors of an orthonormal basis sum to 1
Given that:
$x_1 x_2 + y_1 y_2 + z_1 z_2=0$,
$x_1 x_3 + y_1 y_3 + z_1 z_3=0$,
$x_2 x_3 + y_2 y_3 + z_2 z_3=0$,
$x_1^2+y_1^2+z_1^2=1$,
$x_2^2+y_2^2+z_2^2=1$,
$x_3^2+y_3^2+z_3^2=1$
How to prove ...
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Infinite series of squares of elements of orthonormal basis of $L^2(a,b)$
Let $(e_n)$ be orthonormal basis of $L^2(a,b)$. Prove that, for every set $A \subset (a,b)$ which has positive Lebesgue measure, $\sum_{n=1}^{\infty} \int_A |e_n(x)|^2 dx = \infty.$ Also, prove that $\...
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How do I show $\frac{1}{\sqrt L}e^{\frac{2\pi inx}{L}}$ is an orthonormal set for $L^2[a, b]$ (where $|b-a|=L$)?
Let $$f_n=\frac{1}{\sqrt L}e^{\frac{2\pi inx}{L}}$$
To check that they're orthonormal I calculate their scalar product:
$$(f_n, f_k)=\int_a^b \bar f_n(x)f_k(x)dx$$
Of course if n=k the scalar product ...
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Does the covariant derivative of the unit normal vector field vanish (in this special case)?
Assume we have a Riemannian manifold $(M,g)$, and then define the "cylinder" $M' := M\times\mathbb{R}$. Then $M'$ together with the product metric $g'_p : (v_1,r_1),(v_2,r_2)\mapsto g(v_1,...
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Deriving components of a bra vector using orthonormal basis vectors
I'm stuck already in an introductory chapter about bras and kets. I included all of the context but I'm stuck at figure 1.5.
In figure 1.5, what does $\alpha_j$ mean? In the previous figures, the ...
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Who coined the term Orthonormal? [closed]
Does anyone know who coined the term orthonormal to refer to a basis that is orthogonal and normal.
In such a poorly named mathematical world (looking at you conditionally convergent series) I think ...
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Is this property true for subspaces of a Hilbert Space?
Suppose $W$ is a subspace of a Hiblert space(We can assume separability if it simplifies things) and fix an orthonormal family $\{e_{n}\}$ .(Note that this need not be an orthonormal basis as ...
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$T$ be a finite rank operator of rank $n$. There exist orthonormal set $\{u_1,\ldots,u_n\}$ such that $Th=\sum_{i=1}^n\langle h,u_i\rangle Tu_i$
$T$ be a finite rank operator of rank $n$. There exist orthonormal set $\{u_1,\ldots,u_n\}$ such that $\{Tu_1,\ldots,Tu_n\}$ is linearly independent and $$Th=\sum_{i=1}^n\langle h,u_i\rangle Tu_i$$
...
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