科学与数学setosa.io编辑精选
直观解释特征向量与特征值
Eigenvectors and eigenvalues explained visually
2015 — 2018，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这是 setosa.io 上的一个可交互可视化页面，用拖动向量、观察线性变换后向量是否仍落在原直线上的方式，直观展示特征向量与特征值的几何含义。页面侧重动画演示和亲手操作，文字讲解较为简略，适合已有线性代数基础的人建立直觉。
为什么经久不衰：交互动画让抽象的线性变换变得可玩、可见，短小精悍，是可视化教学早期被反复引用的经典范例。
HN 评论精选
动画出色但文字解释不足，缺少对“不共线时如何”的说明。
漂亮的动画和拖动向量的功能都很好，但很遗憾（我不是在否定作者）这里其实没多少“解释”。第一句只说了特征值的用途，第二句交代示意图的背景，第三句说“如果能做 X，那么 v 是特征向量、λ 是特征值”。可是如果做不到 X 呢？如果 v、(0,0) 和 Av 不共线呢？一个优秀讲解的骨架已经在了，但肉还没长出来，多几句话就能好很多。
discardorama查看英文原文
The pretty animations are nice, and the ability to manipulate the vectors is very nice; however, I am sorry to say (and I do not mean this negatively) that there's not much "explanation". The first sentence just describes the utility of the Eigens (so no explanation there). The next lays out the setting for the diagram. And the third says, "if we can do X, then v is an eigenvector and \lambda an eigenvalue". But... what if you can't do "X" ? What if v, (0,0) and Av are not colinear? The skeleton of a great explanation is there, but the meat isn't there yet. A few more sentences would go a long way in making this better. I appreciate the OP's effort, and I hope this will come across as constructive criticism.
应从一个具体问题出发推导，最后再用图，而非一上来就画图。
图表很美，但我不觉得它能让人真正理解。我会先抛出问题：如何方便地计算相互依赖的量的求和；再展示朴素算法，然后引入矩阵、向量和特征值得到解法，最后才用图形展示各步骤。我惊讶这篇文章没这么做，因为这在我看来是美国讲解科学问题的标准套路。
bsaul查看英文原文
Very beautiful graphs, but i don't think it's going to make people understand anything. I would start with the problem : easily compute sums of dependant values, then show a naive computation, then use matrices, vectors and eigenvalues to come to a solution, and only then, show a graphical representation of the steps performed. I'm surprised that this post isn't following this method, because i've come to think it's the standard way of explaining scientific things in the US.
特征空间和 S1/S2 的表述让人误解，需要更清楚的措辞和区分。
很酷，但作为外行，我对特征空间以及 S1/S2 两条线的描述非常困惑。我猜（理由见下）建议改成：“特征空间是特殊的直线，从线上任意一点出发，特征值映射后仍落回这条线。本例中有两条，标为 S1 和 S2。”并补充说明 S1 像“谷”会吸引附近的点序列，S2 像“山丘”，稍微偏离就会被推得更远。原文先说“与特征向量共线的每个点都是另一个特征向量，那条线就是特征空间”，让我一度以为整个象限都是二维特征空间，可后文又说特征空间是线，前后矛盾。
Terr_查看英文原文
Very cool, but as a layman I was a very confused by the description of eigenspaces and the S1/S2 lines. I'm just guessing here (reasoning below) but I'd like to suggest phrasing like: "Eigenspaces are special lines, where any starting-point along them yields an eigenvalue that lands back on the same line. In these examples two exist, labeled, S1 and S2." "Eigenspaces show where there is 'stability' from repeated applications of the eigenvector. Some act like 'troughs' which attract nearby series of points (S1) while others are like hills (S2) where any point even slightly outside the stable peak yields eigenvalues further away. ______ Original post / detailed-reaction: > First, every point on the same line as an eigenvector is another eigenvector. That line is an eigenspace. At first I though this statement-of-fact meant that the whole tweakable quadrant of the X/Y plot (at a minimum) is an unbroken 2D Eigenspace, because every point within it can be "covered" by a dashed line (a 2D "vector") if I pick the appropriate start-point. However, the last sentence also says eigenspaces are (despite the "space" in their name) lines, which throws the earlier interpretation into doubt. > As you can see below, eigenspaces attract this sequence S1 and S2 were displayed earlier, but not explained, now this section implies that those lines are the Eigenspaces? If so, what is the difference between S1 and S2? Playing with the chart, I assume they are the "forward" and "reverse" for repeat-applications of the transformation.
推荐 3Blue1Brown 系列，认为理解特征值的前提是先掌握线性变换等基础。
3Blue1Brown 的 YouTube 系列很适合建立线性代数直觉。其中较靠后的一集讲特征值/特征向量，大意是：特征相关的东西本身很直接，但只有对前置内容（线性变换、行列式、线性方程组、基变换等）有扎实的直观理解才讲得通；对特征概念的困惑通常源于基础不牢，而不是特征本身。整个系列都用动画展示数值运算对坐标系做了什么。
steamer25查看英文原文
3Blue1Brown has a good series on YouTube for building intuition in linear algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab In one of the last videos in the (relatively short) series, he discusses eigen-*: ~'eigen-stuffs are straight-forward but only make sense if you have a solid visual understanding of the pre-requisites (linear transformations, determinants, linear systems of equations, change of basis, etc.). Confusion about eigen-stuffs usually has more to do with a shaky foundation than the eigen-things themselves' https://youtu.be/PFDu9oVAE-g All of the videos in the series, including this later one on eigen-things, focus on animations to show what the number crunching is doing to the coordinate system.
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