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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
Happy year of the dragon.
Greetings to all MATLAB users,
Although the MATLAB Flipbook contest has concluded, the pursuit of ‘learning while having fun’ continues. I would like to take this opportunity to highlight some recent insightful technical articles from a standout contest participant – Zhaoxu Liu / slandarer.
Zhaoxu has contributed eight informative articles to both the Tips & Tricks and Fun channels in our new Discussions area. His articles offer practical advice on topics such as customizing legends, constructing chord charts, and adding color to axes. Additionally, he has shared engaging content, like using MATLAB to create an interactive dragon that follows your mouse cursor, a nod to the upcoming Year of the Dragon in 2024!
I invite you to explore these articles for both enjoyment and education, and I hope you'll find new techniques to incorporate into your work.
Our community is full of individuals skilled in MATLAB, and we're always eager to learn from one another. Who would you like to see featured next? Or perhaps you have some tips & tricks of your own to contribute. Remember, sharing knowledge is a collaborative effort, as Confucius wisely stated, 'When I walk along with two others, they may serve me as my teachers.'
Let's maintain our commitment to a continuous learning journey. This could be the perfect warm-up for the upcoming 2024 contest.
Can you see them?
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!
One of my colleauges, Michio, recently posted an implementation of Pong Wars in MATLAB
- Here's the code on GitHub.https://lnkd.in/gZG-AsFX
- If you want to open with MATLAB Online, click here https://lnkd.in/gahrTMW5
- He saw this first here: https://lnkd.in/gu_Z-Pks
Making me wonder about variations. What might the resulting patterns look with differing numbers of balls? Different physics etc?
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?
Is there a reason for TMW not to invest in 3D polyshapes? Is the mathematical complexity of having all the same operations in 3D (union, intersection, subtract,...) prohibitive?
Quick answer: Add set(hS,'Color',[0 0.4470 0.7410]) to code line 329 (R2023b).
Explanation: Function corrplot uses functions plotmatrix and lsline. In lsline get(hh(k),'Color') is called in for cycle for each line and scatter object in axes. Inside the corrplot it is also called for all axes, which is slow. However, when you first set the color to any given value, internal optimization makes it much faster. I chose [0 0.4470 0.7410], because it is a default color for plotmatrix and corrplot and this setting doesn't change a behavior of corrplot.
Suggestion for a better solution: Add the line of code set(hS,'Color',[0 0.4470 0.7410]) to the function plotmatrix. This will make not only corrplot faster, but also any other possible combinations of plotmatrix and get functions called like this:
h = plotmatrix(A);
% set(h,'Color',[0 0.4470 0.7410])
for k = 1:length(h(:))
get(h(k),'Color');
end
We are thrilled to announce the grand prize winners of our MATLAB Flipbook contest! This year, we invited the MATLAB Graphics Infrastructure team, renowned for their expertise in exporting and animation workflows, to be our judges. After careful consideration, they have selected the top three winners:
Judge comments: Creative and realistic rendering with well-written code
2nd place - Christmas Tree / Zhaoxu Liu
Judge comments: Festive and advanced animation that is appropriate to the current holiday season.
Judge comments: Nice translation of existing shader logic to MATLAB that produces an advanced and appealing visual effect.
In addition, after validating the votes, we are pleased to announce the top 10 participants on the leaderboard:
- Tim
- Zhaoxu Liu / slandarer
- KARUPPASAMYPANDIYAN M
- Dhimas Mahardika S.Si., M.Mat
- Augusto Mazzei
- Jenny Bosten
- Lucas
- Jr
- Victoria
- ME
Congratulations to all! Your creativity and skills have inspired many of us to explore and learn new skills, and make this contest a big success!
The MATLAB Flipbook Mini Hack contest has concluded! During the 4 weeks, over 600 creative animations have been created. We had a lot of fun and a great learning experience! Thank you, everyone!
Now it’s the time to announce week 4 winners. Note that grand prize winners will be announced shortly after we validate votes on winning entries.
Realism:
Holiday & Season:
Abstract:
Cartoon:
Congratulations, weekly winners!We will reach out to you shortly for your prizes.
Looking for an opportunity to practice your AI skills on a real-world problem? Interested in AI for climage change? Sign up for the Kelp Wanted challenge, which tasks participants with developing an algorithm that can detect the presence of kelp forests from satellite images.
Participants of all skill levels from anywhere in the world are welcome to compete!
MathWorks provides the following resources for all participants:
Have you marveled at the breathtaking, natural-looking animations crafted by the creative minds in the Flipbook Mini Hack contest? Think of @Tim, @Jenny Bosten, and @Zhaoxu Liu / slandarer- their work is nothing short of extraordinary.
So, what's their secret? Adam Danz, a developer in the MATLAB Graphics and Charting team and a top community contributor, has graciously unveiled the mysteries in his latest blog post - "Creating natural textures with power-law noise: clouds, terrains, and more." The post offers simple, step-by-step instructions and code snippets, empowering you to grasp these enchanting techniques effortlessly.
Check it out and we hope it sparks your creativity and serves as a wellspring of inspiration. With only 3 days remaining before the contest draws to a close, it's time to dive into the code and let your imagination soar!
In Week 3, we reached the 400-animations milestone! Let’s work together to achieve the 500-animations goal!
During the last week of the contest, we strongly encourage you to inspire your colleagues, classmates, or friends to vote. Voters will also have the opportunity to win a MATLAB T-shirt.
Mini Hack Winners - Week 3
Math, Physics, or Science explanation:
Most creative remix:
40:
Math is beautiful:
Mashup (Combined themes):
Nature:
Holidays:
Congratulations, winners!
In week 4, we’d love to see more entries in the following categories:
- Holidays:
- Seasons:
- Abstract:
- Mashup (mixed categories)
A gentle reminder that you have a direct impact on the next generation of animation tools in MATLAB! Don’t forget to share your thoughts and ideas with us.
i've finished writing the code for my ~masterpiece~ and it doesn't run on the contest new entry page. it runs on matlab on my desktop and in the matlab live editor in my browser, but not where i really need it to. usually it'll draw the first frame the first time i run the code in a new window, but not any subsequent times. whether i hit "run" or "create animation", the screen grays out as it's supposed to, but then returns to normal without generating the first frame or animation.
i'm not getting a timeout warning or any error messages. i timed the code on my laptop and it takes about 100ish seconds to run and generate the animation the first time i run it before clearing everything from my workspace (and then it takes about twice as long each subsequent time, which makes me a bit nervous. can't figure out why this is the case at all!), which is well within the 255 second limit for the contest. as far as i know, the most computationally expensive function i'm using is patch(). i'm pretty close to the character limit, but i don't know if that's part of the problem.
i tried vectorizing some of the code instead of using for loops, which made the code run slightly slower. i tried using a nested function instead of using drawframe() and an auxiliary function, but that didn't help. i tried clearing all variables except f at the end of drawframe() to no avail. i tried using the close command at the start of drawframe(), with similar success. i updated chrome, closed a bunch of windows, tried safari, used my sister's laptop, all with no luck.
has anyone else had this or a similar problem? any advice?
thanks!
Just in two weeks, we already have 300 entries and 2,000 votes! We are so impressed by your creative styles, artistic talents, and ingenious programming techniques.
Now, it’s time to announce the weekly winners!
Mini Hack Winners - Week 2
Time & Space:
Trees:
Cartoon:
Symbolism:
Realism:
Fractal:
Multi-Entry Stories:
Explanatory:
Light:
MATtropolis:
Congratulations! Each of you won your choice of a T-shirt, a hat, or a coffee mug. We will contact you after the contest ends.
New categories for Week 3!
- 40: MathWorks will celebrate its 40th Anniversary in 2024. You will have opportunities to win special prizes!
- Most creative remix
Categories we’d love to see more entries in
- Multi-Entry Stories (e.g. Jr’s Matzilla series)
- MATropolis
- Math/Physics Explanation
Finally, just a reminder that you have a direct impact on the next generation of animation tools in MATLAB! Don’t forget to share your thoughts and ideas with us.
Dear contest participants,
Happy Friday! I am thrilled to share some exciting news with you. Cleve Moler, the father of MATLAB, has just published a captivating blog post dedicated to our outstanding Flipbook Mini Hack contest!
In his blog post, Cleve expresses his genuine admiration for the multitude of creative styles, artistic talents, and ingenious programming techniques that have been showcased throughout the contest. What's even more remarkable is that he handpicked some of his personal favorites! You don't want to miss this opportunity to see what Cleve has to say.
But that's not all! Your exceptional animations have sparked immense interest among the MathWorks teams. This means you have a direct impact on the next generation of animation tools in MATLAB! We value your thoughts and ideas, so please don't hesitate to share them with us.
Let's keep up the phenomenal work in the remaining two weeks, and together, we'll continue to push the boundaries of creativity and innovation.
Wishing you all an incredible weekend ahead!
Enthusiastically yours,
Chen
I think it would be a really great feature to be able to add an Alpha property to the basic "Line" class in MATLAB plots. I know that I have previously had to resort to using Patch to be able to plot semitransparent lines, but there are also so many other functions that rely on the "Line" class.
For example, if you want to make a scatter plot from a table with things specified into groups, you can use ScatterHistogram or gscatter but since gscatter uses the Line class, you can't adjust the marker transparency. So if you don't want the histograms, you are stuck with manually separating it and using scatter with hold on.
Dedicated to all Flipbook contest lovers! :D
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