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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
I was looking into the possibility of making a spin-to-win prize wheel in MATLAB. I was looking around, and if someone has made one before they haven't shared. A labeled colored spinning wheel, that would slow down and stop (or I would take just stopping) at a random spot each time. I would love any tips or links to helpful resources!
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.
Many of the examples in the MATLAB documentation are extremely high quality articles, often worthy of attention in their own right. Time to start celebrating them! Today's is how to increase Image Resolution using deep learning
https://uk.mathworks.com/help/deeplearning/ug/single-image-super-resolution-using-deep-learning.html
Struct is an easy way to combine different types of variants. But now MATLAB supports classes well, and I think class is always a better alternative than struct. I can't find a single scenario that struct is necessary. There are many shortcomings using structs in a project, e.g. uncontrollable field names, unexamined values, etc. What's your opinion?
I am confused, is the matlab answer better or Julia’s?
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Enjoy yourself and have fun! We're committed to fostering a supportive and educational environment. Dive into discussions, share your expertise, and grow your knowledge. We're excited to see what you'll contribute to the community!
Have you ever used Live Tasks in MATLAB? MathWorks development team would like to get some feedback on your experience – what did you like and not like. Especially, if you know about it but don’t use it frequently, we would like to understand why?
Please tell us what you think by submitting your response to this form https://forms.office.com/r/ui1EGqAFDx
MathWorks just released three new courses on Coursera liseted below. If you work with image or video data and are wanting to incorporate deep learning techniques into your workflow, this is a great opporutnity. The course creators monitor the discussion forums, so you can ask questions and get feedback on your work. Below are links to the three courses and a quick description of a project you'll complete in each.
- Introduction to Computer Vision for Deep Learning. You'll train a classifier to classify images of people signing the American Sign Language alphabet.
- Deep Learning for Object Detection. Move from just classification to finding object locations. You'll train a model to find different types of parking available on the MathWorks campus.
- Advanced Deep Learning Techniques for Computer Vision. You'll train anomaly detection models for medical images and use AI-assisted labeling auto label images.
Can anyone provide insight into the intended difference between Discussions and Answers and what should be posted where?
Just scrolling through Discussions, I saw postst that seem more suitable Answers?
What exactly does Discussions bring to the table that wasn't already brought by Answers?
Maybe this question is more suitable for a Discussion ....
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!
Hello Community!
We are working on a new translation experience for the MathWorks website and products. The goal is to make it easy for people to see content in the best language for them.
Step 1 is learning from those of you who use another language instead of, or in addition to English. If this sounds like you, we'd love your response to a brief survey.
Feel free to comment here as well. Thanks in advance!
We've released an open-source implementation of STIPA (Speech Transmission Index for Public Address) on GitHub!
What is STIPA?
Speech Transmission Index is a metric designed to assess the quality of speech transmission through a communication channel. It quantifies the intelligibility of speech signals based on amplitude modulations, providing a standardized measure crucial for evaluating public address systems and communication equipment. STIPA is a version of STI using a simplified measurement method and only one test signal.
Quality Representation:
STI values range from 0 to 1, categorizing speech transmission quality from bad to excellent. The raw STI score can be transformed into the likelihood of intelligibility of syllables, words, and sentences being comprehended.
Verification Tests:
To ensure reliability, we've conducted verification tests according to the IEC 60286-16 standard. Download the test signals and run the stipaVerificationTests.m script from our GitHub repository.
Control Measurements:
We've performed comparative measurements in a university auditorium, showcasing the validity of our implementation.
License:
Our STIPA implementation is distributed under the GNU General Public License 3, reflecting our commitment to open-source collaboration.
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?
i am just thinking to make a project on software defined ratio SDR using matlab and its toolboxes but I am UG student in ECE don't know how to start can we have discussion here and want the guidance from the best or good persons in the field of wireless communication
Hello, Community Members!
Every day, we witness the incredible exchange of knowledge as over 100,000 users visit our community for answers or to get some code. We have such a vibrant community because of the dedicated group of contributors who volunteer their time and expertise to help one another.
We learned that many community users are looking for different ways to show their appreciation to contributors. In response, we're thrilled to announce the launch of our latest feature – Skill Endorsements.
When you visit a contributor's profile page, you'll notice a brand-new 'Endorsements' tab. Here, you have the power to acknowledge the skills of your fellow members by either endorsing a new skill or bolstering existing ones.
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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?
I noticed a couple new replies show up on the recent poll a day or so ago, but since then, the page can't be loaded anymore in any browser I've tried.
Is MathWorks going to spend 5 years starting in 2024 making Python the #1 supported language?
I'm not sure it's authentic information, and am looking forward to a high level of integration with python.
Reference:
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