Positive Definite Matrices: Definition and Criteria | 正定矩阵：定义与判别方法

📚 Positive Definite Matrices: Definition and Criteria | 正定矩阵：定义与判别方法

A positive definite matrix is one of the most important concepts in linear algebra, with deep connections to quadratic forms, optimization, statistics, and differential equations. This article provides a clear definition and the main criteria for determining whether a real symmetric matrix is positive definite.

正定矩阵是线性代数中最重要的概念之一，与二次型、最优化、统计学和微分方程有着深刻联系。本文给出清晰的定义，并系统介绍判定实对称矩阵是否正定的主要方法。

1. Definition Based on Quadratic Forms | 基于二次型的定义

Let A be a real symmetric matrix of size n × n. We say that A is positive definite if for every nonzero column vector x ∈ ℝⁿ, the quadratic form satisfies

设 A 是一个 n × n 实对称矩阵。若对任意非零列向量 x ∈ ℝⁿ，二次型满足

xᵀ A x > 0

then A is called positive definite. If the inequality is always nonnegative, A is positive semidefinite.

则称 A 为正定矩阵。若不等式恒为非负，则称 A 为半正定矩阵。

2. The Symmetry Assumption | 对称性假设

In most definitions, positive definiteness is only defined for symmetric (or Hermitian) matrices. This is natural because xᵀ A x is a scalar, and its value depends only on the symmetric part (A + Aᵀ)/2. For a real matrix, the quadratic form can be rewritten using its symmetric part.

大多数定义中，正定性仅针对对称（或 Hermitian）矩阵。这是因为 xᵀ A x 是标量，其值只依赖于对称部分 (A + Aᵀ)/2。对于实矩阵，二次型可以用其对称部分重新表示。

In this article, unless otherwise stated, A is assumed to be a real symmetric matrix. Similar results hold for complex Hermitian matrices with x* A x > 0.

在本文中，除非特别说明，均假设 A 是实对称矩阵。对于复 Hermitian 矩阵，类似结果在 x* A x > 0 的意义下成立。

3. Eigenvalue Criterion | 特征值判别法

A symmetric matrix A is positive definite if and only if all of its eigenvalues are strictly positive.

对称矩阵 A 正定当且仅当其所有特征值均为严格正数。

A positive definite ⇔ λᵢ > 0 for all i = 1, …, n

This criterion follows from the Spectral Theorem: if A = QΛQᵀ with orthogonal Q and diagonal Λ, then xᵀ A x = yᵀ Λ y = Σ λᵢ yᵢ², where y = Qᵀ x. Since y is nonzero whenever x is nonzero, the sign of the quadratic form is determined entirely by the eigenvalues.

该判别法源于谱定理：若 A = QΛQᵀ，其中 Q 正交，Λ 对角，则 xᵀ A x = yᵀ Λ y = Σ λᵢ yᵢ²，这里 y = Qᵀ x。由于 x 非零时 y 也非零，二次型的符号完全由特征值决定。

4. Leading Principal Minor Test | 顺序主子式判别法

The leading principal minor test is a purely algebraic method that avoids computing eigenvalues. For an n × n matrix A, the k-th leading principal minor is the determinant of the upper-left k × k submatrix.

顺序主子式判别法是一种纯代数方法，无需计算特征值。对于 n × n 矩阵 A，第 k 个顺序主子式是左上角 k × k 子矩阵的行列式。

Δₖ = det(A[1..k, 1..k]), k = 1, …, n

Then A is positive definite if and only if all leading principal minors are positive:

则 A 正定当且仅当所有顺序主子式均为正：

Δ₁ > 0, Δ₂ > 0, …, Δₙ > 0

For example, for a 2 × 2 matrix, this reduces to a > 0 and ac − b² > 0.

例如，对于 2 × 2 矩阵，条件简化为 a > 0 且 ac − b² > 0。

5. Sylvester’s Criterion | Sylvester 准则

Sylvester’s criterion is exactly the leading principal minor test stated above. It is one of the most widely used tests in practice because determinants are easy to compute for small matrices.

Sylvester 准则正是上述顺序主子式判别法。它是实践中应用最广泛的判别法之一，因为对于小矩阵，行列式计算非常简便。

However, note that Sylvester’s criterion applies only to positive definiteness, not to positive semidefiniteness. For semidefinite matrices, all principal minors (not just leading ones) must be nonnegative, which is a stronger condition.

但要注意，Sylvester 准则仅适用于正定性，不适用于半正定性。对于半正定矩阵，需要所有主子式（不仅仅是顺序主子式）非负，这是一个更强的条件。

6. Pivot Test | 主元判别法

When performing Gaussian elimination without row exchanges, the pivots are the diagonal entries after elimination. A symmetric matrix is positive definite if and only if all pivots are positive.

在不进行行交换的高斯消元中，主元是消元后对角线上的元素。一个对称矩阵正定当且仅当所有主元均为正数。

A = LDLᵀ, with D diagonal and dᵢᵢ > 0 for all i

This is often called the LDLᵀ decomposition. The pivots dᵢᵢ can be computed during elimination; if any is zero or negative, the matrix is not positive definite.

这通常称为 LDLᵀ 分解。主元 dᵢᵢ 可在消元过程中计算；若任一主元为零或负数，则矩阵不是正定的。

7. Cholesky Decomposition | Cholesky 分解

An equivalent condition is the existence of a unique lower triangular matrix L with positive diagonal entries such that

一个等价条件是存在唯一的具有正对角元素的下三角矩阵 L，使得

A = L Lᵀ

This is the Cholesky decomposition. If A is positive definite, the Cholesky factor L is unique and can be computed efficiently. Conversely, if such an L exists, then for any x ≠ 0, xᵀ A x = xᵀLLᵀx = ||Lᵀx||² > 0, so A is positive definite.

这就是 Cholesky 分解。若 A 正定，则 Cholesky 因子 L 唯一存在且可高效计算。反之，若存在这样的 L，则对任意 x ≠ 0，xᵀ A x = xᵀLLᵀx = ||Lᵀx||² > 0，因此 A 正定。

8. Relation to Invertibility | 与可逆性的关系

Every positive definite matrix is invertible, but not every invertible symmetric matrix is positive definite.

每个正定矩阵都是可逆的，但并非每个可逆对称矩阵都是正定的。

Positive definite ⇒ all eigenvalues > 0 ⇒ determinant > 0 ⇒ invertible.

正定 ⇒ 所有特征值 > 0 ⇒ 行列式 > 0 ⇒ 可逆。
The converse fails: for example, diag(−1, −2) has determinant 2 > 0 but is negative definite, not positive definite.

逆命题不成立：例如 diag(−1, −2) 的行列式为 2 > 0，但它是负定矩阵，不是正定矩阵。

9. Negative Definite and Semidefinite Cases | 负定与半正定情形

Closely related concepts are used in optimization and multivariable calculus:

在优化和多元微积分中，密切相关的重要概念包括：

Condition	Quadratic form	Eigenvalues
Positive definite	xᵀ A x > 0 for x ≠ 0	All λᵢ > 0
Positive semidefinite	xᵀ A x ≥ 0 for all x	All λᵢ ≥ 0
Negative definite	xᵀ A x < 0 for x ≠ 0	All λᵢ < 0
Negative semidefinite	xᵀ A x ≤ 0 for all x	All λᵢ ≤ 0
Indefinite	xᵀ A x takes both signs	Some λᵢ > 0, some λᵢ < 0

In particular, the Hessian matrix at a critical point determines the nature of a local extremum: positive definite Hessian gives a local minimum, negative definite gives a local maximum.

特别地，临界点处的 Hessian 矩阵决定了局部极值的性质：正定 Hessian 给出局部极小值，负定 Hessian 给出局部极大值。

10. Worked Example | 实例演示

Consider the matrix

考虑矩阵

A = [ 2 -1 0 ; -1 2 -1 ; 0 -1 2 ]

Check positive definiteness using three methods.

我们用三种方法判别其正定性。

Method 1: Eigenvalues. Solving det(A − λI) = 0 gives λ = 2, 2 ± √2. Since all are positive, A is positive definite.

方法一：特征值。由 det(A − λI) = 0 解得 λ = 2, 2 ± √2。全部为正，故 A 正定。

Method 2: Leading principal minors. Δ₁ = 2 > 0; Δ₂ = 2 × 2 − (−1)² = 3 > 0; Δ₃ = det(A) = 4 > 0. Hence A is positive definite.

方法二：顺序主子式。Δ₁ = 2 > 0；Δ₂ = 2 × 2 − (−1)² = 3 > 0；Δ₃ = det(A) = 4 > 0。因此 A 正定。

Method 3: Cholesky decomposition. We can find L = [[√2, 0, 0], [−1/√2, √(3/2), 0], [0, −√(2/3), √(4/3)]]. Since L exists with positive diagonal entries, A is positive definite.

方法三：Cholesky 分解。可求得 L = [[√2, 0, 0], [−1/√2, √(3/2), 0], [0, −√(2/3), √(4/3)]]。由于 L 存在且对角元为正，故 A 正定。

11. Common Mistakes and Pitfalls | 常见错误与陷阱

Students often confuse the leading principal minor test with the condition “all principal minors positive”. For positive definiteness, the leading principal minors suffice. However, for positive semidefiniteness, requiring only leading principal minors nonnegative is not sufficient.

学生经常混淆“顺序主子式判别法”和“所有主子式为正”的条件。对于正定性，顺序主子式已足够；但对于半正定性，仅要求顺序主子式非负并不充分。

Example: A = [[0,0],[0,−1]] has leading principal minors 0 and 0, but it is not positive semidefinite because xᵀ A x = −y² < 0 for x = (0,1).

例：A = [[0,0],[0,−1]] 的顺序主子式分别为 0 和 0，但它不是半正定的，因为取 x = (0,1) 时 xᵀ A x = −y² < 0。
Also, a matrix with positive elements on the diagonal is not necessarily positive definite. The off-diagonal entries play a crucial role.

此外，对角线元素全为正的矩阵不一定是正定的，非对角线元素起着关键作用。

12. Summary | 总结

To determine whether a real symmetric matrix is positive definite, you can use any of the following equivalent conditions:

要判断一个实对称矩阵是否正定，可以使用以下任意等价条件：

xᵀ A x > 0 for all x ≠ 0

All eigenvalues are strictly positive.

所有特征值均严格为正。
All leading principal minors are positive (Sylvester’s criterion).

所有顺序主子式均为正（Sylvester 准则）。
All pivots from Gaussian elimination are positive.

高斯消元得到的所有主元均为正。
There exists a unique lower triangular matrix L with positive diagonal entries such that A = LLᵀ.

存在唯一的下三角矩阵 L，其对角元素为正，且 A = LLᵀ。

Mastering these criteria is essential for IB Higher Level Mathematics, as they appear in questions involving quadratic forms, eigenvalues, and optimization. Practice each method and recognize which one is most efficient for a given matrix.

掌握这些判别方法对于 IB 高级数学至关重要，它们在涉及二次型、特征值和最优化的问题中经常出现。通过练习熟悉每种方法，并学会在给定矩阵时选择最高效的判别方式。

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Positive Definite Matrices: Definition and Criteria | 正定矩阵：定义与判别方法