Questions tagged [inner-products]
For questions about inner products and inner product spaces, including questions about the dot product.
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Characterizing the dual lattice of an real vector space
Let $n\in\mathbf{N}$, $V=$ {$(x_1,...,x_n)\in\mathbf{R}^n|\sum\limits_{i=1}^n x_i=0$} be a real vector space, Let $l=V\cap\mathbf{Z}^n$ in $V$. Let $(\cdot,\cdot)$ be a restriction to $V$ of the ...
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Finding an explicit formula for a norm induced by an inner product
$\langle \cdot, \cdot \rangle$ is an inner product defined on $\mathbb{R}^2$ and $\| \cdot \|$ is the norm induced by it. The norm satisfies the following conditions:
$$
\sup_{\boldsymbol{x} \in \...
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Is Hermiticity dependent on the choice of inner product? [duplicate]
We have the two standard definitions of Hermiticity:
$A_{ij} = \overline{A_{ji}}$ (i.e. $A = \overline{A^T}$), which references the basis
$\langle A x, y\rangle = \langle x, Ay\rangle$ (i.e. $A = A^\...
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Complex conjugates in orthonormal basis.
Okay, hello, I am not from the algebra world so I'm sorry for any mistakes. So, assume that we have given some orthonormal vectors $e_1,e_2,\ldots,e_n$ in the matrix $A$. We now consider a complex ...
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Maximum number of pairs of vectors such that $\langle v_i , v_j \rangle \neq 0 $
Let $\langle \cdot , \cdot \rangle $ denote the standard inner product on $\mathbb{R}^7$ . Let $\Sigma =\{ v_1 , v_2 , ..., v_5 \} \subseteq \mathbb{R}^7 $ be a set of unit vectors such that $\langle ...
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Question about orthogonal eigenvectors
I know that given a normal matrix A, then distinct eigenvalues correspond to orthogonal eigenvectors. I've seen the proof by contradiction which only relies on the definition of the inner product. ...
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How to get Angles using Dot Product
In the below link , I have a picture with two 3d Models numbered 1 and 2 . I am trying to figure out the angle between the Base of the Part and the Side Flange(face) of the Part. As inputs, I have ...
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Visuallizing complex vectors?
I'm learning Linear algebra course, and i was stumbled with an expamle that showing how the complex vectors are perpendicular when the dot product equals to zero.
The question was to find the dot ...
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Ask an identity on inner products [closed]
Question: how to get the the identity
$$
\left\|y_n-y_m\right\|^2=2\left\|x-y_n\right\|^2+2\left\|x-y_m\right\|^2-4\left\|x-\frac{y_n+y_m}{2}\right\|^2
$$
The background of the question is from the ...
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Why I can use in my theorem that the norm $\|\cdot\|$ in an inner product space is strictly convex?
I am wondering why in my post If $K$ is a closed convex subset of an real inner product space $X$ that is contained in a complete subset of $X$, then $K$ is Chebyshev. I was able to use the following:
...
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Let $C$ be a convex cone in this space, $z \in C$, and $K = C + z$.
Let $(X, \langle \cdot, \cdot \rangle)$ be a vector space over $\mathbb{R}$ with an inner product. Let $C$ be a convex cone in this space, $z \in C$, and $K = C + z$. If $x \in X$ and $y_0 \in K$, ...
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Linear Algebra Done Right (4th edition) - Problem 23 of section 6A [duplicate]
I'm struggling with the problem 23 of section 6A from the 4th edition of Axler's "Linear Algebra Done Right":
Suppose $v_1,\ldots,v_m \in V$ are such that $\|v\|\leq 1$ for each $k=1,\ldots,...
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Prove that for any $x \in X$, $ \|x - y\|^2 \geq \|x - P_K(x)\|^2 + \|y - P_K(x)\|^2 $ for every $y \in K$.
I want to prove the following two theorems:
Let $K$ be a convex subset of the real inner product space $X$, $x \in X$, and $y_0 \in K$. Then $y_0 = P_K(x)$ if and only if
\begin{equation}
\langle ...
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Every closed convex subset of a finite-dimensional subspace of real inner product space is Chebyshev.
In my post I was now able to prove the following theorem : If $K$ is a closed convex subset of an real inner product space $X$ that is contained in a complete subset of $X$, then $K$ is Chebyshev.. ...
user1348309
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If $K$ is a closed convex subset of an real inner product space $X$ that is contained in a complete subset of $X$, then $K$ is Chebyshev.
If $K$ is a closed convex subset of an real inner product space $X$ that is contained in a complete subset of $X$, then $K$ is Chebyshev.
Note: $P_K(x) := \{ y \in K \mid \|x - y\| = d(x, K) \}$. $K$ ...
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Separable Hilbert spaces, total orthonormal sets and Schauder basis in Banach spaces
A Schauder basis for a Banach space $X$ is called the sequece $\{e_n\}$ where $x\in X$ can be expanded as:
$$x=\sum^\infty_{k=1} a_ke_k$$
A total orthonormal basis in a Hilbert space $H$ is a Schauder ...
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The inner product of $xf'(x)$ and $g(x)$, $\int x f'(x) g(x)$, always vanishes if $f(x)$ and $g(x)$ are probability distribution functions
Let $f(x)$ and $g(x)$ be probability distribution functions defined in the interval $[a,b]$, and assume that the mean of the distribution $g$ exists, i.e. $\int_a^b x g(x) = \mu_g$, and the derivative ...
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Let $\{p_k\} \subset P_n \subset C_2[a,b]$ and suppose $\|p_k\| \rightarrow 0$. Show that $p_k \rightarrow 0$ uniformly on $[a,b]$.
Task: Let $\{p_k\} \subset P_n \subset C_2[a,b]$ and suppose $\|p_k\| \rightarrow 0$. Show that $p_k \rightarrow 0$ uniformly on $[a,b]$. That is, for each $\epsilon > 0$, there exists an integer $...
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I would like to prove that the corollary also implies the theorem
Theorem: Every closed convex set in a Hilbert space is approximatively Chebyshev compact.
Proof: Let $K$ be a closed convex set in a Hilbert space $H$. Choose an arbitrary Cauchy sequence $(y_n)$ in $...
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Let $K$ be a convex set, $x \in K$, and let $\{y_n\}$ in $K$ be a minimizing sequence for $x$. Show that $\{y_n\}$ is a Cauchy sequence.
Let $X$ be a real inner product space and let $K$ be a complete convex set. Fix any $x \in X$. Suppose that $\{y_n\}$ is minimizing for $x$: $\|x - y_n\| \rightarrow d(x, K)$. I will prove that the ...
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If two infinite dimensional vector spaces are isomorphic, does an inner product isomorphism exist between them? [closed]
The following proposition is true:
If $V$ and $W$ are finite dimensional vector spaces over a field $\mathbb{F}$, the following is equivalent.
$\quad$ (1) $V$ and $W$ are isomorphic as inner product ...
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Then every approximately compact set in this space is proximal.
Let $(X, \langle \cdot, \cdot \rangle)$ be a vector space over $\mathbb{R}$ with an inner product. Then every approximately compact set in this space is proximal. Now, I want to find an example where ...
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Representations on inner product spaces
Let $V$ be a $\mathbb{F}$-inner product space and $\rho: G \to GL(V)$ a linear representation that preserves the inner product. If $\mathbb{F}=\mathbb{C}$, then we say $V$ is a unitary representation, ...
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If the set $K$ is convex in this space, then the sets $Cl(K)$ and $int(K)$ are also convex in this space. [duplicate]
Task: Let $(X, \langle \cdot, \cdot \rangle)$ be a vector space over $\mathbb{R}$ with an inner product. If the set $K$ is convex in this space, then the sets $Cl(K)$ and $int(K)$ are also convex in ...
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Signature of a metric induced from scalar product space
Suppose I have a scalar product space $\mathbb{R}^n_k$, that is, $\mathbb{R}^n$ equipped with a scalar product $\langle \ , \ \rangle$ such that:
$$\langle x,y \rangle = -\sum^{k}_{i=1} x_iy_i + \sum^{...
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if $\lambda \in [0,1]$ and $x' = \lambda x + (1 - \lambda)P_K(x)$, then $P_K(x') = P_K(x)$
I have problem proving following theorem: If $K$ is a convex Chebyshev set in an inner product space $X$ and $x \in X \setminus K$, show:
if $\lambda \in [0,1]$ and $x' = \lambda x + (1 - \lambda)P_K(...
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If $K$ is a convex set in this space, then every element $x \in X$ has at most one best approximation in $K$.
I would like to prove the following theorem: Let $(X, \langle \cdot, \cdot \rangle)$ be a vector space over $\mathbb{R}$ with an inner product. If $K$ is a convex set in this space, then every element ...
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The set $A$ is convex in this space if and only if $co(A) = A$.
First, let me state the following definitions:
Let $(X, \langle \cdot, \cdot \rangle)$ be a vector space over $\mathbb{R}$ with a scalar product. A set $K$ in this space is called convex if $\lambda ...
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uniqueness of a projection of a vector
Let $W$ a $k$-dimentional subspace of a finite dimensional inner product vector space $V$. If $v\in V$, then it is well known that the vector
$$
p=\langle v,w_1\rangle w_1+\ldots+\langle v,w_k\rangle ...
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Is there a coherent notion of "weak orthogonality" of functions/distributions?
Recently I gave an answer to Laplace transforms of non-exponential, non-sinusoidal functions that claims the following:
Our [change-of-basis] interpretation doesn't quite survive [the generalization ...
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Semi definite inner product is either positive or negative inner product
According to nlab's definition of inner product
( see : https://ncatlab.org/nlab/show/inner+product+space ),
inner product is defined on vector space over field with involution. And it satisfies :
$\...
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Prove that a space is a Hilbert Space with inner product different to the usual inner product used
I'm working with following academic paper : Stability of the solutions of differential equations whose author is Bernard Beauzamy.
I'm trying prove that the two next spaces $\mathcal{B}_2$ y $P_2$ are ...
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What is an inner product in terms of abstract algebra? [closed]
I'm trying to apply my knowledge of linear algebra to reinforce my understanding of general abstract algebra.
Since the output of a inner product is in a different form than the inputs (scalar vs ...
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Why weak convergence doesn't Imply strong convergence on $\infty$- dimensional Hilbert spaces.
To start off, I know this is wrong. Im hoping someone can explain to me where Im going wrong. I know that on a finite dimensional Hilbert space that weak convergence implies strong convergence. But I ...
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Inner Product of Matrices based on the determinant
Usually the inner product of matrices $A,B\in\mathbb{C}^{m\times n}$ is defined to be $\left\langle A,B \right\rangle = \text{tr}(A^\dagger B)$, also know as the Frobenius Inner Product, and this does ...
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Complex functions and inner product $\langle \frac{\partial f}{\partial z} , g\rangle $
I'm working through this academic paper : stability of the soljutions of differential equations whose author is Bernard Beauzamy. A link to paper
In the academic paper, it work with the next norm
\...
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Difficulty understanding the meaning of "volume" with the scalar triple product
I am computing the scalar triple product of two vectors a, b, and their orthogonal cross product c. A special case of this is where a x b = c, which leads to the scalar triple product equalling 0.
My ...
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Normal operator with certain condition is self adjoint
Let $V$ be a finite dimensional complex vector space with inner product, and let $T\in L(V)$ be a normal linear operator such that $T^9=T^8$. Show that $T$ is a self-adjoint projection.
The condition ...
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Alternative to pairwise summation for aggregating scalar products of multiple vectors
There is a problem involving the aggregation of high-dimensional vectors in a way that avoids the computational cost of pairwise summation. Specifically, each data point is represented by a 16-...
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The inner product on a vector space whose field is R
if we can make different definitions of the inner product in a vector space does that mean that the length and angles we get from each can serve different purposes and return different values , ...
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When does an inner product on $C(K)$ come from integration?
Let $K$ be a compact Hausdorff space and suppose $\langle \cdot, \cdot \rangle$ is an inner product on $C(K)$ such that $\langle f, g \rangle \ge 0$ whenever $f(t),g(t)\ge 0$ for all $t \in K$. It ...
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Proof that the Dual Transformation of a Dual Transformation is Itself
Let $A$ be an invertible linear transformation in $\mathbb{R}^{n\times n}$. Then, we know that $A$ takes $\mathbb{R}^{n-1}$ hyperplanes in its domain to $\mathbb{R}^{n-1}$ hyperplanes in its image. ...
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Dot Product Intuition help
I'm in a quest for understanding the dot product. I thought of the same base case here, but the proof that by rotating both vectors won't change the dot product isn't intuitive, does anyone have any ...
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Proving the projection $f \mapsto \sum_{i=1}^k \langle f, e_i\rangle e_i$ is continuous
Let $f: [0, T] \rightarrow L^2(\mathbb{R}^n)$ and $\{e_i\}$ a basis for $L^2(\mathbb{R}^n)$. I am following a proof where the authors would like to show that the finite projections
$$P_kf := \sum_{i=1}...
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How axioms of inner product ensure that an instantiation/realization capture notion of angle correctly?
Axiomatic definition of inner product can lead to various instantiations like euclidean inner product or complex inner product or weighted inner product etc.
Whatever the special case, we can be sure, ...
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Show that function is positive definite
In the process of showing that the function\begin{equation}
\langle \mathbf{z}, \mathbf{w}\rangle=\overline{z_1}w_1+(1+i)\overline{z_1}w_2+(1-i)\overline{z_2}w_1+3\overline{z_2}w_2
\end{equation} is ...
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Can one solve this complex linear equation
Suppose the equations are as follows:
$$\text{tr}\{AH\}= c$$
where $c \in \mathbb{C}$ and $A$ is a symmetric matrix in $M_{N}(\mathbb{R})$ are both known and $H$ is a Hermitian matrix which looks like:...
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Interpretation of $\max \sup$
I am reading the regret analysis proof of LinUCB given in Lattimore's Bandit Algorithms. He makes the following assumption:
$$ \max\limits_{t\in[n]}\sup\limits_{a,b\in\mathcal{A}_t} \langle\theta^* , ...
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Why do we have 2 types of products for vectors?
Why is it that we have two types of products for vectors ? Why do we have one scalar product and one vector product? If we were concerned with the magnitude of the product, wouldn't it be easier to ...
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Jacobian and Gradient map multiplication vs dot product confusion in model
Hi I have a very simple model and I'm trying to learn the math of it.
Basically, I have an input matrix X (m x n). An output matrix Y (m x n) is formed from some convolution H. The figure of merit is ...
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