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eye(3) - diag(ones(1,3))
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3009 票
lazymatlab
lazymatlab
Last activity 2024 年 2 月 20 日

행렬식은 면적이다? Determinant is area?

2 x 2 행렬의 행렬식은
  • 행렬의 두 row 벡터로 정의되는 평행사변형의 면적입니다.
  • 물론 두 column 벡터로 정의되는 평행사변형의 면적이기도 합니다.
  • 좀 더 정확히는 signed area입니다. 면적이 음수가 될 수도 있다는 뜻이죠.
  • 행렬의 두 행(또는 두 열)을 맞바꾸면 행렬식의 부호도 바뀌고 면적의 부호도 바뀌어야합니다.
일반적으로 n x n 행렬의 행렬식은
  • 각 row 벡터(또는 각 column 벡터)로 정의되는 N차원 공간의 평행면체(?)의 signed area입니다.
  • 제대로 이해하려면 대수학의 개념을 많이 가지고 와야 하는데 자세한 설명은 생략합니다.(=저도 모른다는 뜻)
  • 더 자세히 알고 싶으시면 수학하는 만화의 '넓이 이야기' 편을 추천합니다.
  • 수학적인 정의를 알고 싶으시면 위키피디아를 보시면 됩니다.
  • 이렇게 생겼습니다. 좀 무섭습니다.
아래 코드는...
  • 2 x 2 행렬에 대해서 이것을 수식 없이 그림만으로 증명하는 과정입니다.
  • gif 생성에는 ScreenToGif를 사용했습니다. (gif 만들기엔 이게 킹왕짱인듯)
Determinant of 2 x 2 matrix is...
  • An area of a parallelogram defined by two row vectors.
  • Of course, same one defined by two column vectors.
  • Precisely, a signed area, which means area can be negative.
  • If two rows (or columns) are swapped, both the sign of determinant and area change.
More generally, determinant of n x n matrix is...
  • Signed area of parallelepiped defined by rows (or columns) of the matrix in n-dim space.
  • For a full understanding, a lot of concepts from abstract algebra should be brought, which I will not write here. (Cuz I don't know them.)
  • For a mathematical definition of determinant, visit wikipedia.
  • A little scary, isn't it?
The code below is...
  • A process to prove the equality of the determinant of 2 x 2 matrix and the area of parallelogram.
  • ScreenToGif is used to generate gif animation (which is, to me, the easiest way to make gif).
% 두 점 (a, b), (c, d)의 좌표
a = 4;
b = 1;
c = 1;
d = 3;
% patch 색 pre-define
lightgreen = [144, 238, 144]/255;
lightblue = [169, 190, 228]/255;
lightorange = [247, 195, 160]/255;
% animation params.
anim_Nsteps = 30;
% create window
figure('WindowStyle','docked')
ax = axes;
ax.XAxisLocation = 'origin';
ax.YAxisLocation = 'origin';
ax.XTick = [];
ax.YTick = [];
hold on
ax.XLim = [-.4, a+c+1];
ax.YLim = [-.4, b+d+1];
% create ad-bc patch
area = patch([0, a, a+c, c], [0, b, b+d, d], lightgreen);
p_ab = plot(a, b, 'ko', 'MarkerFaceColor', 'k');
p_cd = plot(c, d, 'ko', 'MarkerFaceColor', 'k');
p_ab.UserData = text(a+0.1, b, '(a, b)', 'FontSize',16);
p_cd.UserData = text(c+0.1, d-0.2, '(c, d)', 'FontSize',16);
area.UserData = text((a+c)/2-0.5, (b+d)/2, 'ad-bc', 'FontSize', 18);
pause
%% Is this really ad-bc?
area.UserData.String = 'ad-bc...?';
pause
%% fade out ad-bc
fadeinout(area, 0)
area.UserData.Visible = 'off';
pause
%% fade in ad block
rect_ad = patch([0, a, a, 0], [0, 0, d, d], lightblue, 'EdgeAlpha', 0, 'FaceAlpha', 0);
uistack(rect_ad, 'bottom');
fadeinout(rect_ad, 1, t_pause=0.003)
draw_gridline(rect_ad, ["23", "34"])
rect_ad.UserData = text(mean(rect_ad.XData), mean(rect_ad.YData), 'ad', 'FontSize', 20, 'HorizontalAlignment', 'center');
pause
%% fade-in bc block
rect_bc = patch([0, c, c, 0], [0, 0, b, b], lightorange, 'EdgeAlpha', 0, 'FaceAlpha', 0);
fadeinout(rect_bc, 1, t_pause=0.0035)
draw_gridline(rect_bc, ["23", "34"])
rect_bc.UserData = text(b/2, c/2, 'bc', 'FontSize', 20, 'HorizontalAlignment', 'center');
pause
%% slide ad block
patch_slide(rect_ad, ...
[0, 0, 0, 0], [0, b, b, 0], t_pause=0.004)
draw_gridline(rect_ad, ["12", "34"])
pause
%% slide ad block
patch_slide(rect_ad, ...
[0, 0, d/(d/c-b/a), d/(d/c-b/a)],...
[0, 0, b/a*d/(d/c-b/a), b/a*d/(d/c-b/a)], t_pause=0.004)
draw_gridline(rect_ad, ["14", "23"])
pause
%% slide bc block
uistack(p_cd, 'top')
patch_slide(rect_bc, ...
[0, 0, 0, 0], [d, d, d, d], t_pause=0.004)
draw_gridline(rect_bc, "34")
pause
%% slide bc block
patch_slide(rect_bc, ...
[0, 0, a, a], [0, 0, 0, 0], t_pause=0.004)
draw_gridline(rect_bc, "23")
pause
%% slide bc block
patch_slide(rect_bc, ...
[d/(d/c-b/a), 0, 0, d/(d/c-b/a)], ...
[b/a*d/(d/c-b/a), 0, 0, b/a*d/(d/c-b/a)], t_pause=0.004)
pause
%% finalize: fade out ad, bc, and fade in ad-bc
rect_ad.UserData.Visible = 'off';
rect_bc.UserData.Visible = 'off';
fadeinout([rect_ad, rect_bc, area], [0, 0, 1])
area.UserData.String = 'ad-bc';
area.UserData.Visible = 'on';
%% functions
function fadeinout(objs, inout, options)
arguments
objs
inout % 1이면 fade-in, 0이면 fade-out
options.anim_Nsteps = 30
options.t_pause = 0.003
end
for alpha = linspace(0, 1, options.anim_Nsteps)
for i = 1:length(objs)
switch objs(i).Type
case 'patch'
objs(i).FaceAlpha = (inout(i)==1)*alpha + (inout(i)==0)*(1-alpha);
objs(i).EdgeAlpha = (inout(i)==1)*alpha + (inout(i)==0)*(1-alpha);
case 'constantline'
objs(i).Alpha = (inout(i)==1)*alpha + (inout(i)==0)*(1-alpha);
end
pause(options.t_pause)
end
end
end
function patch_slide(obj, x_dist, y_dist, options)
arguments
obj
x_dist
y_dist
options.anim_Nsteps = 30
options.t_pause = 0.003
end
dx = x_dist/options.anim_Nsteps;
dy = y_dist/options.anim_Nsteps;
for i=1:options.anim_Nsteps
obj.XData = obj.XData + dx(:);
obj.YData = obj.YData + dy(:);
obj.UserData.Position(1) = mean(obj.XData);
obj.UserData.Position(2) = mean(obj.YData);
pause(options.t_pause)
end
end
function draw_gridline(patch, where)
ax = patch.Parent;
for i=1:length(where)
v1 = str2double(where{i}(1));
v2 = str2double(where{i}(2));
x1 = patch.XData(v1);
x2 = patch.XData(v2);
y1 = patch.YData(v1);
y2 = patch.YData(v2);
if x1==x2
xline(x1, 'k--')
else
fplot(@(x) (y2-y1)/(x2-x1)*(x-x1)+y1, [ax.XLim(1), ax.XLim(2)], 'k--')
end
end
end
Chen Lin
Chen Lin
Last activity 2024 年 2 月 14 日

Happy Valentine's Day

See code here in our community contest area. (author: @Zhaoxu Liu / slandarer)
And what do you do for Valentine's Day?
K
K
Last activity 2024 年 2 月 10 日

contact technical support

which technical support should I contact/ask for the published Simscape example?
Chen Lin
Chen Lin
Last activity 2024 年 2 月 10 日

Fun Spring Festival Couplets

Happy year of the dragon.

Chen Lin
Chen Lin
Last activity 2024 年 2 月 15 日

How many circles do you find in this picture?

Can you see them?
Hannah
Hannah
Last activity 2024 年 2 月 8 日

Flipbook Mini Hack

I have been procrastinating on schoolwork by looking at all the amazing designs created in the last MATLAB Flipbook Mini Hack! They are just amazing. The voting is over but what are y'all's personal favorites? Mine is the flapping butterfly, it is for sure a creation I plan to share with others in the future!
Chen Lin
Chen Lin
Last activity 2024 年 2 月 1 日

Video Resources - How to get started with MatGPT

I recently discovered a 2-minute video that introduces MatGPT, and I believe it's a resource worth sharing. The creator highlights MatGPT's impressive capabilities by demonstrating how it tackles the classic Travelling Salesman Problem.
With more than 13,000 downloads on File Exchange, MatGPT is gaining traction among users. I strongly recommend taking it for a spin to experience its potential firsthand.
K
K
Last activity 2024 年 1 月 31 日

how accurate are the answers of the AI Playground

how accurate are the answers of the AI Playground regarding information that are not specifiyed in the documentation?
Mike Croucher
Mike Croucher
Last activity 2024 年 1 月 29 日

Pong Wars in MATLAB

One of my colleauges, Michio, recently posted an implementation of Pong Wars in MATLAB
Making me wonder about variations. What might the resulting patterns look with differing numbers of balls? Different physics etc?
We're thrilled to announce the roll-out of some new features that are going to supercharge your Playground experience! Here's what's new:
Copy/Download code from the script area
You can now effortlessly Copy/Download code from the script area with just a single click. Copy code or Download your script directly as .m files and keep your work organized and portable.We hope this will allow you to effortlessly transfer your work from Playground to MATLAB Desktop/Online.
Run Code directly from the Chat panel
Execute code snippets from the chat section with a single click. This new affordance means saving a step since you no longer have to insert code and then hit run from the toolstrip to execute instead just hit run in the chat panel to see the output immediately in the script area
Enhanced visual Experience
Customize your Playground workspace by expanding or collapsing the chat and script sections. Focus on what matters most to you, whether it's AI chat or working on your script.
We hope you will love these updates. Try them out and let us know your feedback.
Mike Croucher
Mike Croucher
Last activity 2024 年 10 月 22 日

Multiple AIs in the chat playground

When I want to understand a problem, I'll often use different sources. I'll read different textbooks, blog posts, research papers and ask the same question to different people. The differences in the solutions are almost always illuminating.
I feel the same way about AIs. Sometimes, I don't want to ask *THE* AI...I want to ask a bunch of them. They'll have different strengths and weaknesses..different personalities if you want to think of it that way.
I've been playing with the AI chat arena and there really is a lot of difference between the answers returned by different models. https://lmarena.ai/?arena
I think it would be great if the MATLAB Chat playgroundwere to allow the user to change which AI they were talking with.
What does everyone else think?
NODIRABDUSHAKHIDOV
NODIRABDUSHAKHIDOV
Last activity 2024 年 4 月 1 日

help, how can i use this AI?

how can i use this AI?
We are excited to unveil the ‘Open in MATLAB Online from File Exchange’ feature, which offers MATLAB users a new way to open File Exchange content!
Previously, to experiment with File Exchange code, you were required to download the file and execute it in MATLAB. But now, there's a quicker and easier way to explore the code!
You will find the ‘Open in MATLAB Online’ button next to the ‘Download’ button (see the screenshot below). A simple click transports you directly into the MATLAB Online workflow. It's that straightforward and effortless.
We strongly encourage you to try this new feature. Please share your questions, comments, or ideas by responding to this post!
I have been finding the AI Chat Playground very useful for daily MATLAB use. In particular it has been very useful for me in basically replacing or supplementing dives into MATLAB documentation. The documentation for MATLAB is in my experience uniformly excellent and thorough but it is sometimes lengthy and hard to parse and the AI Chat is a great one stop shop for many questions I have. However, I would find it very useful if the AI Chat could answer my queries and then also supply a link directly to the documentation. E.g. a box at the bottom of the answer that is basically
"Here is the documentation on the functions AI Chat referred to in this response"
could be neat.
I recently wrote about the new ODE solution framework in MATLAB over the The MATLAB Blog The new solution framework for Ordinary Differential Equations (ODEs) in MATLAB R2023b » The MATLAB Blog - MATLAB & Simulink (mathworks.com)
This was a very popular post at the time - many thousands of views. Clearly everyone cares about ODEs in MATLAB.
This made me wonder. If you could wave a magic wand, what ODE functionality would you have next and why?
Hi
I am using simulink for the frequency response analysis of the three phase induction motor stator winding.
The problem is that i can't optimise the pramaeter values manually, for this i have to use genetic algrothem. But iam stucked how to use genetic algorithum to optimise my circuit paramter values like RLC. Any guidence will be highly appreciated.
sky
sky
Last activity 2024 年 1 月 18 日

Relative pose in 2D: problem 1

I'm having problem in its test 6 ... passing 5/6 what would be the real issue..
am wring Transformation matrix correct.. as question said SSW should be 202.5 degree...
so what is the issue..
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