λ 2×2 矩阵特征值与特征向量Eigenvalue & Eigenvector Calculator (2×2)

使用说明与常见问题How-to & FAQ

「2×2 矩阵特征值与特征向量」是什么？What is the Eigenvalue & Eigenvector Calculator (2×2) tool?

解 det(A - λI) = 0 得特征值（含复数），并给出对应的特征向量。本地处理。可设置「a₁₁」、「a₁₂」、「a₂₁」、「a₂₂」。Solve det(A - λI) = 0 for the eigenvalues (complex included) and give eigenvectors. Processed locally. Settings: a₁₁, a₁₂, a₂₁, a₂₂.

如何使用「2×2 矩阵特征值与特征向量」？How do I use the Eigenvalue & Eigenvector Calculator (2×2) tool?

填写 2×2 矩阵的四个元素 a₁₁、a₁₂、a₂₁、a₂₂，即可得到特征值（包括复数）和对应的特征向量。结果会随输入实时更新。Enter the four entries a₁₁, a₁₂, a₂₁ and a₂₂ of the 2×2 matrix to get its eigenvalues (complex ones included) and eigenvectors. The result updates as you type.

更大矩阵？Bigger matrices?

本页只算 2×2；更大的矩阵需要数值迭代算法。2×2 only; larger matrices need iterative numerical methods.

能举个计算实例吗？Can you show a worked example?

例如在「2×2 矩阵特征值与特征向量」中填 a₁₁ = 2，a₁₂ = 1，a₂₁ = 1，a₂₂ = 2，结果区显示：迹 / 行列式：4 / 3；λ₁：3；v₁：[0.707107, 0.707107]；λ₂：1；v₂：[0.707107, -0.707107]For example, entering a₁₁ = 2, a₁₂ = 1, a₂₁ = 1, a₂₂ = 2 in the Eigenvalue & Eigenvector Calculator (2×2) gives: Trace / determinant: 4 / 3; λ₁: 3; v₁: [0.707107, 0.707107]; λ₂: 1; v₂: [0.707107, -0.707107]

哪些输入会被检查？Which inputs are validated?

「2×2 矩阵特征值与特征向量」会检查这些输入——「a₁₁」、「a₁₂」、「a₂₁」、「a₂₂」：必须填写；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Eigenvalue & Eigenvector Calculator (2×2) checks its inputs — "a₁₁", "a₁₂", "a₂₁", "a₂₂": required; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「2×2 矩阵特征值与特征向量」的设置：「a₁₁」默认 2；「a₁₂」默认 1；「a₂₁」默认 1；「a₂₂」默认 2。每次生成结果都按当前设置处理。Settings in the Eigenvalue & Eigenvector Calculator (2×2) — "a₁₁" default 2; "a₁₂" default 1; "a₂₁" default 1; "a₂₂" default 2. The result always uses the current settings.

计算精度如何？How precise are the results?

「2×2 矩阵特征值与特征向量」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Eigenvalue & Eigenvector Calculator (2×2) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「2×2 矩阵特征值与特征向量」免费吗？需要注册吗？Is the Eigenvalue & Eigenvector Calculator (2×2) free? Do I need an account?

「2×2 矩阵特征值与特征向量」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Eigenvalue & Eigenvector Calculator (2×2) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.