📚 Positive-definite Matrices | 正定矩阵
Positive-definite matrices are a special class of symmetric matrices that play a crucial role in many areas of mathematics, including optimization, statistics, and machine learning. Understanding their properties and how to test for positive definiteness is essential for IB Mathematics students, particularly when studying quadratic forms, Hessian matrices, and extrema of multivariable functions. This article explores the definition, criteria, and applications of positive-definite matrices in a comprehensive yet exam-focused manner.
正定矩阵是一类特殊的对称矩阵，在优化、统计学和机器学习等许多数学领域中起着关键作用。理解它们的性质以及如何检验正定性对于 IB 数学学生来说至关重要，尤其是在学习二次型、黑塞矩阵和多元函数极值时。本文以综合但紧扣考试的方式探讨正定矩阵的定义、判别法及其应用。
1. Definition and Basic Properties | 定义与基本性质
trix A of size n × n is said to be positive definite if for every non-zero column vector x ∈ ℝⁿ, the scalar xᵀ A x is strictly positive. That is, xᵀ ensures that the quadratic form associated with A is always positive except at the origin.
一个 n × n 的实对称矩阵 A 被称为正定的，如果对于每个非零列向量 x ∈ ℝⁿ，标量 xᵀ A x 严格大于零。即对于所有 x ≠ 0，有 xᵀ A x > 0。该条件确保与 A 相关的二次型在原点之外始终为正。
From this definition, it follows that all diagonal entries of a positive-definite matrix must be positive (set x to be the i-th standard basis vector). Moreover, a positive-definite matrix is always invertible since its determinant is positive.
从这一定义可推出，正定矩阵的所有对角线元素必须为正（令 x 为第 i 个标准基向量）。此外，正定矩阵总是可逆的，因为其行列式为正。
2. Quadratic Forms | 二次型
c form in n variables is a homogeneous degree-2 polynomial that can be written as Q(x) = xᵀ or all x ≠ 0, then A is positive definite. The quadratic form expands to ∑ᵢ aᵢᵢ xᵢ² + 2∑ᵢ<ⱼ aᵢⱼ xᵢ xⱼ.
n 个变量的二次型是一个齐次二次多项式，可以写成 Q(x) = xᵀ A x，其中 A 是对称矩阵。Q(x) 的符号决定了矩阵的类型。如果对于所有 x ≠ 0 有 Q(x) > 0，则 A 是正定的。二次型展开式为 ∑ᵢ aᵢᵢ xᵢ² + 2∑ᵢ<ⱼ aᵢⱼ xᵢ xⱼ。
boloid (or ellipsoid in 2D) centered at the origin, with a unique global minimum at x = 0.
从几何角度看，正定二次型定义了一个开口向上的抛物体（在二维中为椭球面），以原点为中心，在 x = 0 处有唯一的全局最小值。
3. Eigenvalue Criterion | 特征值判别法
A symmetric matrix A is positive definite if and only if all its eigenvalues are strictly positive. Since real symmetric matrices have real eigenvalues, we can compute them and check their signs. This is one of the most powerful and theoretically fundamental criteria.
对称矩阵 A 是正定的当且仅当其所有特征值都严格为正。由于实对称矩阵具有实特征值，我们可以计算它们并检查符号。这是最有力且理论最基础的判别法之一。
Proof sketch: Using the spectral theorem, A = Q Λ Qᵀ, where Q is orthogonal and Λ diagonal with eigenvalues. Then xᵀ A x = (Qᵀ x)ᵀ Λ (Qᵀ x) = ∑ λᵢ yᵢ², which is positive for all non-zero y (hence x) iff every λᵢ > 0.
证明概要：利用谱定理，A = Q Λ Qᵀ，其中 Q 是正交矩阵，Λ 是特征值对角阵。那么 xᵀ A x = (Qᵀ x)ᵀ Λ (Qᵀ x) = ∑ λᵢ yᵢ²，该式对所有非零 y (因此也所有非零 x)为正当且仅当每个 λᵢ > 0。
4. Sylvester's Criterion | 西尔维斯特准则（主子式判别法）
Sylvester' e determinant of the upper-left k × k submatrix. For n=2, we check a₁₁ > 0 and det(A) > 0.
西尔维斯特准则提供了一种无需计算特征值的实用检验方法：对称矩阵正定当且仅当其所有顺序主子式为正。第 k 个顺序主子式是左上角 k × k 子矩阵的行列式。对于 n=2，我们检查 a₁₁ > 0 且 det(A) > 0。
is often simpler than solving the characteristic polynomial, especially for 3×3 matrices.
例子：A = [[2, -1], [-1, 2]]。这里 a₁₁=2>0 且 det(A)=4−1=3>0，所以 A 是正定的。使用顺序主子式通常比解特征多项式更简单，尤其对于 3×3 矩阵。
It is important to note that only leading principal minors need to be positive; checking all principal minors (not just leading) is necessary for positive semidefiniteness.
需要注意的是只有顺序主子式需要为正；检查所有主子式（不仅是顺序的）是半正定性的必要条件。
5. Positive Semidefinite Matrices | 半正定矩阵
A symmetric matrix is positive semidefinite if xᵀ t leading) non-negative. Semidefinite matrices arise in constrained optimization and covariance matrices.
对称矩阵半正定是指对所有 x 有 xᵀ A x ≥ 0。这意味着特征值 ≥ 0 且所有主子式（不仅是顺序的）非负。半正定矩阵出现在约束优化和协方差矩阵中。
and det=0, but the second-order leading minor is zero, so it fails positive definiteness.
例如，A = [[1, 2], [2, 4]] 的特征值为 0 和 5，因此它是半正定但非正定的。顺序主子式为 1>0 和 det=0，但二阶顺序主子式为零，因此不满足正定性。
6. Symmetry Requirement | 对称性要求
In the context of quadratic forms, only the symmetric part of a matrix matters because xᵀ A x = xᵀ ((A+Aᵀ is symmetric. The standard definitions used in IB assume symmetry unless stated otherwise.
在二次型的背景下，只有矩阵的对称部分有意义，因为 xᵀ A x = xᵀ ((A+Aᵀ)/2) x。因此，当我们说一个矩阵是正定的，几乎总是暗含它是对称的。IB 中使用的标准定义除非另有说明都假设对称性。
For non-symmetric matrices, the condition xᵀ A x > 0 can still hold but the eigenvalues might not be real. However, in IB exams, positive-definite matrices are always symmetric real matrices.
对于非对称矩阵，条件 xᵀ A x > 0 可能仍然成立，但特征值可能不是实数。然而在 IB 考试中，正定矩阵总是实对称矩阵。
7. Cholesky Decomposition | 楚列斯基分解
Every positive-definite matrix A has a unique Cholesky decomposition A = L Lᵀ ctorization is extremely useful for solving linear systems and in Monte Carlo simulations.
每个正定矩阵 A 都有唯一的楚列斯基分解 A = L Lᵀ，其中 L 是具有正对角线元素的下三角矩阵。该分解对于求解线性方程组和蒙特卡洛模拟非常有用。
For instance, A = [[4, 2], [2, 3]] yields L = [[2, 0], [1, √2]]. The existence of Cholesky factorization is an equivalent definition of positive definiteness for symmetric matrices.
例如，A = [[4, 2], [2, 3]] 可分解为 L = [[2, 0], [1, √2]]。楚列斯基分解的存在性是对称矩阵正定的一个等价定义。
8. Connection to Inner Products | 与内积的联系
A positive-definite matrix defines an inner product on ℝⁿ via ⟨x, y⟩ = xᵀ y because A is symmetric and positive definite. This generalizes the standard dot product.
正定矩阵通过 ⟨x, y⟩ = xᵀ A y 在 ℝⁿ 上定义了一个内积。内积的公理（对称性、线性性、正定性）恰好因为 A 是对称正定的而得到满足。这推广了标准点积。
matrices to geometry, allowing notions of length and angle based on A. The associated norm ‖x‖_A = √xᵀ A x is strictly positive for non-zero x.
这一视角将正定矩阵与几何联系起来，允许基于 A 的长度和角度概念。相关的范数 ‖x‖_A = √xᵀ A x 对于非零 x 严格为正。
9. Application in Optimization | 在优化中的应用
In multivariable calculus, the Hessian matrix of second partial derivatives at a critical point is used to classify extrema. If the Hessian is positive definite, the function has a local minimum. This is the second derivative test extended to n dimensions.
在多元微积分中，临界点处的二阶偏导数黑塞矩阵用于分类极值。如果黑塞矩阵是正定的，则函数有一个局部极小值。这是二阶导数检验向 n 维的推广。
sufficient condition for a local minimum is H positive definite, i.e., fₓₓ > 0 and det(H) > 0.
对于标量函数 f(x, y)，黑塞矩阵 H = [[fₓₓ, fₓᵧ], [fᵧₓ, fᵧᵧ]]。局部极小值的充分条件是 H 正定，即 fₓₓ > 0 且 det(H) > 0。
This reasoning is foundational in unconstrained optimization and machine learning algorithms such as Newton's method and trust-region methods.
这一推理是无约束优化和机器学习算法（如牛顿法和信赖域方法）的基础。
10. Worked Examples | 实例演练
ing principal minors: 2>0, det(A)=4−1=3>0 → yes. Eigenvalues are 1 and 3 → all positive. xᵀ 2x₁² + 2x₂² + 2x₁x₂ which can be rewritten as ½[(x₁+x₂)² + 3(x₁−x₂)²] > 0 for non-zero x.
例1：判断 A = [[2, 1], [1, 2]] 是否正定。顺序主子式：2>0，det(A)=4−1=3>0 → 是。特征值为 1 和 3 → 全部为正。xᵀ A x = 2x₁² + 2x₂² + 2x₁x₂ 可重写为 ½[(x₁+x₂)² + 3(x₁−x₂)²] > 0 对于非零 x。
Example 2: For what value of c is A = [[1, c], [c, 4]] positive definite? Conditions: 1>0 (ok), det = 4 − c² > 0 ⇒ |c| < 2. Also check diagonal entries are positive (satisfied). So c ∈ (−2, 2).
例2：c 取何值时 A = [[1, c], [c, 4]] 正定？条件：1>0（成立），det = 4 − c² > 0 ⇒ |c| < 2。同时检查对角线元素为正（满足）。所以 c ∈ (−2, 2)。
11. Negative Definite and Indefinite Matrices | 负定矩阵与不定矩阵
A symmetric matrix is negative definite if xᵀ gative definite matrix alternate in sign, starting with a₁₁ < 0, then det of 2×2 > 0, etc.
对称矩阵是负定的，如果对所有 x ≠ 0 有 xᵀ A x < 0，这等价于 −A 是正定的。负定矩阵的顺序主子式符号交替变化，以 a₁₁ < 0 开始，然后 2×2 行列式 > 0，等等。
en the quadratic form takes both signs. For instance, A = [[1, 0], [0, -1]] is indefinite.
如果一个对称矩阵既有正特征值又有负特征值，则称为不定矩阵。此时
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