Eigenvalues of Matrices: Definition and Computation | 矩阵特征值的定义与计算方法

📚 Eigenvalues of Matrices: Definition and Computation | 矩阵特征值的定义与计算方法

differential equations, Markov chains, and graph theory. In this article, we will define eigenvalues precisely, derive the characteristic equation, and practise effective computational methods for 2×2 and 3×3 matrices.

特征值是线性代数中最核心的概念之一，贯穿 IB 数学分析与方法 HL 课程中矩阵部分的内容。特征值揭示了矩阵的基本结构，并为对角化、微分方程组、马尔可夫链和图论等强大应用奠定基础。本文将精确定义特征值，推导特征方程，并系统练习 2×2 与 3×3 矩阵的特征值计算方法。

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1. What is an Eigenvalue? | 什么是特征值？

When a matrix A multiplies a vector v, the result Av is generally a new vector with a different direction and length. However, for some special non-zero vectors, the product Av is simply a scalar multiple of v: the direction stays the same, and only the length changes. These special vectors are called eigenvectors, and the corresponding scalar factor is called an eigenvalue.

当矩阵 A 乘以向量 v 时，结果 Av 通常是一个方向和长度都不同的新向量。然而，对于某些特殊的非零向量，乘积 Av 恰好是 v 的标量倍数：方向不变，只有长度改变。这些特殊向量称为特征向量，对应的标量因子称为特征值。

th eigenvalue 2. This geometric intuition is the starting point for the formal definition.

例如，设矩阵 A = [[2, 0], [0, 3]]。取 v = [1, 0]，则 Av = [2, 0] = 2v，因此 v 是特征向量，对应特征值为 2。这种几何直观是正式定义的出发点。

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2. Formal Definition and the Eigenvalue Equation | 正式定义与特征值方程

Let A be an n×n matrix. A scalar λ is an eigenvalue of A if there exists a non-zero vector v in ℝⁿ such that the eigenvalue equation holds:

设 A 是一个 n×n 矩阵。若存在非零向量 v ∈ ℝⁿ 使得以下特征值方程成立，则标量 λ 称为 A 的特征值：

Av = λv, v ≠ 0

The non-zero vector v is called an eigenvector : if v were zero, the equation 0 = 0 would hold for every λ, which carries no information.

非零向量 v 称为对应于 λ 的特征向量。条件 v ≠ 0 至关重要：若 v 为零向量，则等式 0 = 0 对任意 λ 都成立，不包含任何信息。

tity matrix. For this homogeneous system to have a non-trivial solution, the matrix A − λI must be singular, which means its determinant must be zero:

将特征值方程移项可得 (A − λI)v = 0，其中 I 是单位矩阵。为使该齐次方程组存在非零解，矩阵 A − λI 必须是奇异的，即其行列式必为零：

det(A − λI) = 0

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3. The Characteristic Equation | 特征方程

The equation det(A − λI) = 0 is called the characteristic equation of A. For an n×n matrix, the left-hand side is a polynomial of degree n in λ, known as the characteristic polynomial. The eigenvalues of A are exactly the roots of this polynomial.

方程 det(A − λI) = 0 称为 A 的特征方程。对于 n×n 矩阵，其左侧是关于 λ 的 n 次多项式，称为特征多项式。A 的特征值恰好是该多项式的根。

For a general 2×2 matrix A = [[a, b], [c, d]], we compute:

对于一般的 2×2 矩阵 A = [[a, b], [c, d]]，我们计算：

A − λI = [[a−λ, b], [c, d−λ]]

det(A − λI) = (a−λ)(d−λ) − bc = λ² − (a+d)λ + (ad − bc)

Thus the characteristic equation of a 2×2 matrix takes the elegant form:

因此，2×2 矩阵的特征方程可写成简洁的形式：

λ² − tr(A)λ + det(A) = 0

where tr(A) = a + d is the trace (sum of diagonal entries) and det(A) = ad − bc is the determinant. This shortcut saves time and is strongly recommended in exams.

其中 tr(A) = a + d 是矩阵的迹（对角线元素之和），det(A) = ad − bc 是矩阵的行列式。这个捷径可以节省时间，是考试中强烈推荐的方法。

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4. Computing Eigenvalues of 2×2 Matrices | 2×2 矩阵的特征值计算

trix can be summarised in three steps. We will illustrate each step with a worked example.

求 2×2 矩阵特征值的过程可以归结为三个步骤。我们将通过一个例题逐步说明。

Step 1: Form the matrix A − λI. Subtract λ from each diagonal entry, leaving the off-diagonal entries unchanged.

步骤 1：构造矩阵 A − λI。从每个对角线元素中减去 λ，非对角线元素保持不变。

Step 2: Compute its determinant. ultiply the two diagonal entries and subtract the product of the two off-diagonal entries.

步骤 2：计算其行列式。对于 2×2 矩阵，将两个对角线元素相乘，再减去两个非对角线元素的乘积。

Step 3: Set the determinant equal to zero and solve. The resulting quadratic equation is solved either by factorisation or by the quadratic formula.

步骤 3：令行列式等于零并求解。所得二次方程可通过因式分解或求根公式求解。

Worked Example. Find the eigenvalues of A = [[4, 1], [2, 3]].

例题。求 A = [[4, 1], [2, 3]] 的特征值。

Following Step 1, we write A − λI = [[4−λ, 1], [2, 3−λ]]. By Step 2, the determinant is:

按照步骤 1，写出 A − λI = [[4−λ, 1], [2, 3−λ]]。由步骤 2，行列式为：

det(A − λI) = (4−λ)(3−λ) − 1×2 = 12 − 7λ + λ² − 2 = λ² − 7λ + 10

shortcut directly: tr(A) = 7 and det(A) = 10, giving exactly the same quadratic equation.

令行列式等于零，得 λ² − 7λ + 10 = 0。因式分解：(λ − 5)(λ − 2) = 0，因此特征值为 λ = 5 与 λ = 2。注意我们也可以直接使用捷径：tr(A) = 7，det(A) = 10，得到完全相同的二次方程。

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5. Computing Eigenvalues of 3×3 Matrices | 3×3 矩阵的特征值计算

es, but the determinant expansion requires more care. We use cofactor expansion along a convenient row or column.

对于 3×3 矩阵，基本思路相同，但行列式展开需要更加小心。我们沿选定的行或列进行余子式展开。

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