Dtjytyk

Is it a Cyclops number? "Nobody" knows!

What is better: yes / no radio, or simple checkbox?

hyperref and beamer: Meta Data | 'pdftitle' not Working

The (Easy) Road to Code

Does unused member variable take up memory?

Where do you go through passport control when transiting through another Schengen airport on your way out of the Schengen area?

Why are special aircraft used for the carriers in the United States Navy?

3.5% Interest Student Loan or use all of my savings on Tuition?

Is this Paypal Github SDK reference really a dangerous site?

The need of reserving one's ability in job interviews

Did Amazon pay $0 in taxes last year?

What is the oldest European royal house?

When to use the term transposed instead of modulation?

New invention compresses matter to produce energy? or other items? (Short Story)

Does the US political system, in principle, allow for a no-party system?

An Undercover Army

ESPP--any reason not to go all in?

Do natural melee weapons (from racial traits) trigger Improved Divine Smite?

TikZ rotated text looks washed

Are wave equations equivalent to Maxwell's equations in free space?

Should I use HTTPS on a domain that will only be used for redirection?

Naming Characters after Friends/Family

School performs periodic password audits. Is my password compromised?

PTIJ: Aliyot for the deceased

Is the product of two “sample” matrices a sample matrix?

Endomorphisms of a finite dimensional vector spaceUnderstand a matrix expressionWhen is a given matrix-valued function the Jacobian of something?Theorem 3.6 in Sec. 3.9 in Apostol's Calculus vol. 2, 2nd edition: (How) does such a matrix exist?How to sample a singular matrix-variate normal distribution?Describing the space of matrices which “jordanize” a given matrixWhat Is This Matrix Operation Called And What Is It Used For?Orthogonal matrices and inner product definitionIn what conditon, Kronecker product returns linearly independent vectors?The relationship between vector space dualization and matrix transposition.

0

$begingroup$

Let $f,g colon mathbb R^2 to mathbb R$ be two smooth functions.

Take a uniform, square grid of the unit cube in $mathbb R^2$ and let ${p_{ij}}_{i,j=1, ldots, L} subset mathbb R^2$ be the points of the grid. Let $f_{i,j} = f(p_{i,j})$ and $g_{i,j} = g(p_{i,j})$. Consider the matrices $A = (f_{ij})_{i,j=1, ldots, L}$ and $B = (g_{ij})_{i,j=1, ldots, L}$.

Finally, let us agree that we call a $L times L$ matrix $C$ a sample matrix if there exists a function $h$ as above such that $c_{ij} = h(p_{ij})$ for every $i,j$. In particular, $A$ is the sample matrix of $f$ and $B$ of $g$.

Q. Is the matrix product $D := AB$ a sample matrix?

The elements are $d_{ij} := sum_l f_{il} g^{lj}$ and this seems a close relative of the convolution $f ast g$ (or of the Riemann-Stiltjes integral of $f$ in $dg$). However the matrix $D$ cannot be the sample matrix of $f ast g$ (because the convolution is a commutative operation, while matrix product is not).

linear-algebra matrices multivariable-calculus approximation numerical-calculus

share|cite|improve this question

edited 20 hours ago

Romeo

asked Feb 13 at 18:29

RomeoRomeo

3,05721148

$endgroup$

add a comment | 

0

$begingroup$

Let $f,g colon mathbb R^2 to mathbb R$ be two smooth functions.

Take a uniform, square grid of the unit cube in $mathbb R^2$ and let ${p_{ij}}_{i,j=1, ldots, L} subset mathbb R^2$ be the points of the grid. Let $f_{i,j} = f(p_{i,j})$ and $g_{i,j} = g(p_{i,j})$. Consider the matrices $A = (f_{ij})_{i,j=1, ldots, L}$ and $B = (g_{ij})_{i,j=1, ldots, L}$.

Finally, let us agree that we call a $L times L$ matrix $C$ a sample matrix if there exists a function $h$ as above such that $c_{ij} = h(p_{ij})$ for every $i,j$. In particular, $A$ is the sample matrix of $f$ and $B$ of $g$.

Q. Is the matrix product $D := AB$ a sample matrix?

The elements are $d_{ij} := sum_l f_{il} g^{lj}$ and this seems a close relative of the convolution $f ast g$ (or of the Riemann-Stiltjes integral of $f$ in $dg$). However the matrix $D$ cannot be the sample matrix of $f ast g$ (because the convolution is a commutative operation, while matrix product is not).

linear-algebra matrices multivariable-calculus approximation numerical-calculus

share|cite|improve this question

edited 20 hours ago

Romeo

asked Feb 13 at 18:29

RomeoRomeo

3,05721148

$endgroup$

add a comment | 

$begingroup$

Let $f,g colon mathbb R^2 to mathbb R$ be two smooth functions.

Take a uniform, square grid of the unit cube in $mathbb R^2$ and let ${p_{ij}}_{i,j=1, ldots, L} subset mathbb R^2$ be the points of the grid. Let $f_{i,j} = f(p_{i,j})$ and $g_{i,j} = g(p_{i,j})$. Consider the matrices $A = (f_{ij})_{i,j=1, ldots, L}$ and $B = (g_{ij})_{i,j=1, ldots, L}$.

Finally, let us agree that we call a $L times L$ matrix $C$ a sample matrix if there exists a function $h$ as above such that $c_{ij} = h(p_{ij})$ for every $i,j$. In particular, $A$ is the sample matrix of $f$ and $B$ of $g$.

Q. Is the matrix product $D := AB$ a sample matrix?

The elements are $d_{ij} := sum_l f_{il} g^{lj}$ and this seems a close relative of the convolution $f ast g$ (or of the Riemann-Stiltjes integral of $f$ in $dg$). However the matrix $D$ cannot be the sample matrix of $f ast g$ (because the convolution is a commutative operation, while matrix product is not).

linear-algebra matrices multivariable-calculus approximation numerical-calculus

share|cite|improve this question

edited 20 hours ago

Romeo

asked Feb 13 at 18:29

RomeoRomeo

3,05721148

$endgroup$

share|cite|improve this question

edited 20 hours ago

Romeo

asked Feb 13 at 18:29

RomeoRomeo

3,05721148

share|cite|improve this question

edited 20 hours ago

Romeo

asked Feb 13 at 18:29

RomeoRomeo

3,05721148

asked Feb 13 at 18:29

RomeoRomeo

3,05721148

asked Feb 13 at 18:29

RomeoRomeo

3,05721148