📚 Quadratic Forms | 二次型
A quadratic form is a powerful algebraic tool that connects symmetric matrices, eigenvalues, and geometry. In IB Higher Level Mathematics, quadratic forms appear when analysing conic sections, optimising functions, and diagonalising matrices. Mastering them gives you a deeper understanding of linear algebra and its real-world applications.
二次型是一个强大的代数工具，它把对称矩阵、特征值和几何紧密联系在一起。在 IB 高阶数学中，二次型出现在分析圆锥曲线、函数优化以及矩阵对角化等情境里。掌握这一内容能让你更深刻地理解线性代数及其实际应用。
1. What is a Quadratic Form? | 什么是二次型？
f degree two, with no linear or constant parts. In general, we write Q(x₁, ..., xₙ) = Σᵢ Σⱼ aᵢⱼ xᵢ xⱼ, where aᵢⱼ are coefficients, usually real numbers.
n 个变量 x₁, x₂, ..., xₙ 的二次型是一个二次齐次多项式，即每一项的次数恰好为二，没有一次项或常数项。一般形式写作 Q(x₁, ..., xₙ) = Σᵢ Σⱼ aᵢⱼ xᵢ xⱼ，其中系数 aᵢⱼ 一般为实数。
ce the cross term 3xy involves both x and y, and all terms are of degree two. This is exactly the kind of expression we want to study.
对于两个变量，典型的二次型如 Q(x, y) = 2x² + 3xy + 4y²。注意交叉项 3xy 同时含有 x 和 y，且所有项的次数均为二。这正是我们要研究的表达式类型。
Any quadratic form can be expressed compactly using matrices: Q(x) = xT A x, where x is a column vector of variables and A is a symmetric matrix. The symmetry of A ensures the representation is unique and allows us to use powerful theorems from linear algebra.
任何二次型都可以用矩阵形式简洁地表示：Q(x) = xT A x，其中 x 为变量构成的列向量，而 A 是一个对称矩阵。A 的对称性保证了这种表示是唯一的，并使我们能够运用线性代数中的重要定理。
2. Matrix Representation and Symmetry | 矩阵表示与对称性
To construct the symmetric matrix A from Q(x) = Σ aᵢⱼ xᵢ xⱼ we set the diagonal entries Aᵢᵢ equal to the coefficients of xᵢ². For the cross term xᵢ xⱼ (with i ≠ j), we split the coefficient evenly: assign half to Aᵢⱼ and half to Aⱼᵢ. This guarantees A is symmetric.
要从 Q(x) = Σ aᵢⱼ xᵢ xⱼ 构造对称矩阵 A，我们令对角线元素 Aᵢᵢ 等于 xᵢ² 的系数。对于交叉项 xᵢ xⱼ（i ≠ j），我们将系数平分：一半分配给 Aᵢⱼ，另一半分配给 Aⱼᵢ。这样做可以确保 A 是对称的。
1.5yx. Then Q = [x y] A [x; y] indeed yields 2x² + 1.5xy + 1.5yx + 4y² = 2x² + 3xy + 4y².
例如，二次型 Q(x, y) = 2x² + 3xy + 4y² 对应的对称矩阵是 A = [[2, 1.5], [1.5, 4]]，因为 3xy = 1.5xy + 1.5yx。那么 Q = [x y] A [x; y] 确实得到 2x² + 1.5xy + 1.5yx + 4y² = 2x² + 3xy + 4y²。
tries equal half the corresponding cross-term coefficient. This symmetry is not artificial – it comes from the fact that xᵢ xⱼ = xⱼ xᵢ, so we can distribute the coefficient without changing the polynomial.
务必检查非对角线元素是否等于相应交叉项系数的一半。这种对称性并非人为设定，而是源于 xᵢ xⱼ = xⱼ xᵢ，因此我们可以平分系数而不改变多项式本身。
3. The Symmetric Matrix behind Quadratic Forms | 二次型背后的对称矩阵
Every quadratic form can be linked to a unique symmetric matrix. This is crucial because symmetric matrices have real eigenvalues and are orthogonally diagonalisable. The study of quadratic forms thus reduces to the study of symmetric matrices.
每一个二次型都与一个唯一的对称矩阵关联。这一点至关重要，因为对称矩阵具有实特征值且可正交对角化。因此，对二次型的研究就归结为对对称矩阵的研究。
e variables x, y, z, we obtain a 3×3 symmetric matrix. The quadratic form is then Q(x) = xT A x, where x = (x, y, z)T. The symmetry condition Aᵢⱼ = Aⱼᵢ must hold.
矩阵的阶数等于变量的个数。对于三个变量 x, y, z，我们得到一个 3×3 的对称矩阵。二次型即 Q(x) = xT A x，其中 x = (x, y, z)T。必须满足对称条件 Aᵢⱼ = Aⱼᵢ。
In IB exams, you will often be asked to write down the symmetric matrix of a given quadratic form or to reconstruct the quadratic form from a symmetric matrix. Identify the diagonal entries from the squared terms and the off-diagonal entries from half the cross-term coefficients.
在 IB 考试中，你常常需要根据已知二次型写出其对称矩阵，或者由对称矩阵还原二次型。记住：平方项的系数放在对角线，交叉项系数的一半放在对应的非对角线位置。
4. Rank of a Quadratic Form | 二次型的秩
The rank of a quadratic form is defined as the rank of its associated symmetric matrix A. The rank tells us the number of independent linear combinations of variables that actually appear in the form after a suitable change of coordinates.
二次型的秩定义为其对应对称矩阵 A 的秩。秩告诉我们，在适当的坐标变换后，实际出现的变量线性无关组合的个数。
If the matrix A has full rank (equal to the number of variables), the quadratic form is said to be non-degenerate. If the rank is less than the number of variables, some variables can be eliminated through a linear substitution, and the form is degenerate.
如果矩阵 A 是满秩的（等于变量个数），则称该二次型是非退化的。若秩小于变量个数，则某些变量可以通过线性替换消去，此时的二次型是退化的。
1]], whose rank is 1 (not 2). This is degenerate and can be rewritten as (x+y)², showing only one independent 'direction' matters.
例如，二次型 Q(x, y) = x² + 2xy + y² 的矩阵为 [[1,1],[1,1]]，其秩为 1（而非 2）。这是一个退化的二次型，可以改写为 (x+y)²，表明只有一个独立的“方向”起作用。
5. Definiteness: Positive, Negative, Indefinite | 定性：正定、负定与不定
x. Negative definite and negative semi-definite are defined analogously with < 0 and ≤ 0.
如果对所有非零向量 x 都有 Q(x) > 0，则称二次型 Q(x) 是正定的。如果对所有 x 有 Q(x) ≥ 0，且存在某个非零 x 使等式成立，则称为半正定。负定与半负定的定义类似，分别要求 < 0 和 ≤ 0。
- y² is indefinite because it is positive along the x-axis and negative along the y-axis.
若二次型依据 x 的取值既可取正值也可取负值，则称为不定的。例如，Q(x, y) = x² - y² 就是不定二次型，因为它沿着 x 轴为正，沿着 y 轴为负。
Determining the definiteness is important in optimisation: a positive definite quadratic form guarantees a strict local minimum, while a negative definite one guarantees a strict local maximum.
判断定号性在优化中很重要：正定二次型保证函数有严格局部极小值，负定二次型则保证严格局部极大值。
6. Tests for Definiteness | 判定正定性的方法
Eigenvalue test: all eigenvalues are ≥ 0. Similarly, negative definite means all eigenvalues are negative.
特征值检验：对称矩阵 A 是正定的，当且仅当其所有特征值均严格大于零。它是半正定的，当所有特征值 ≥ 0。类似地，负定意味着所有特征值均为负。
Principal minor test (Sylvester's criterion): ipal minors Δ₁, Δ₂, ..., Δₙ (determinants of top-left submatrices). A is positive definite ⇔ all Δₖ > 0. This test is extremely useful for hand calculations with 2×2 or 3×3 matrices.
主子式检验（Sylvester判据）：对于一个 n×n 矩阵，计算其顺序主子式 Δ₁, Δ₂, ..., Δₙ（即左上角各阶子矩阵的行列式）。A 正定 ⇔ 所有 Δₖ > 0。该方法在 2×2 或 3×3 矩阵的手工计算中极为有用。
[b, c]], it is positive definite iff a > 0 and ac - b² > 0. It is negative definite iff a < 0 and ac - b² > 0. If ac - b² < 0, the form is indefinite.
对于 2×2 对称矩阵 A = [[a, b], [b, c]]，它是正定的当且仅当 a > 0 且 ac - b² > 0。它是负定的当且仅当 a < 0 且 ac - b² > 0。若 ac - b² < 0，则二次型不定。
7. Reducing a Quadratic Form: Completing the Square | 化简二次型：配方法
One of the most direct ways to simplify a quadratic form is completing the square. By grouping terms and writing them as perfect squares, we can express a quadratic form as a sum (or difference) of squares, revealing its definiteness and rank.
化简二次型最直接的方法之一是配方法。通过将各项分组并写成完全平方的形式，我们可以把二次型表示为平方和（或平方差），从而揭示其定号性和秩。
then combine with cy². The result is a combination of squares: a(x + (b/a)y)² + (c - b²/a)y².
对于两个变量的二次型 Q(x, y) = ax² + 2bxy + cy²，提公因子 a（若 a ≠ 0），写作 ax² + 2bxy = a(x + (b/a)y)² - (b²/a)y²，再与 cy² 合并。结果是平方组合：a(x + (b/a)y)² + (c - b²/a)y²。
n first complete the square using the y-term, or perform a simple rotation of axes. In IB problems, the method of completing the square is often tested before moving to eigenvalues.
若系数 a = 0 但 b ≠ 0，可以先用 y 项配方，或者进行简单的轴旋转。在 IB 考题中，配方法经常在特征值方法之前被考查。
8. Orthogonal Diagonalisation and the Principal Axes Theorem | 正交对角化与主轴定理
A more systematic method to reduce a quadratic form is orthogonal diagonalisation of the symmetric matrix A. Since A is symmetric, there exists an orthogonal matrix P such that PT A P = D, where D is a diagonal matrix of eigenvalues λ₁, λ₂, ..., λₙ.
一种更系统化的二次型化简方法是对称矩阵 A 的正交对角化。由于 A 是对称的，存在正交矩阵 P 使得 PT A P = D，其中 D 是由特征值 λ₁, λ₂, ..., λₙ 构成的对角矩阵。
Define new variables y = PT x, or equivalently x = P y. Substituting into Q(x) = xT A x gives Q = (P y)T A (P y) = yT (PT A P) y = yT D y = λ₁ y₁² + λ₂ y₂² + ... + λₙ yₙ².
定义新变量 y = PT x，或者等价地 x = P y。代入 Q(x) = xT A x 得到 Q = (P y)T A (P y) = yT (PT A P) y = yT D y = λ₁ y₁² + λ₂ y₂² + ... + λₙ yₙ²。
This is known as the principal axes theorem: any quadratic form can be written as a weighted sum of squares in an orthogonal coordinate system. The coefficients are precisely the eigenvalues. The principal axes are the orthonormal eigenvectors of A.
这就是主轴定理：任一二次型都可以在一个正交坐标系下表示为一组平方项的加权和。系数恰好就是特征值。主轴即 A 的标准正交特征向量。
9. Applications: Conic Sections and Optimisation | 应用：圆锥曲线与优化
. The quadratic part ax² + 2bxy + cy² determines the type of conic (ellipse, parabola, hyperbola) once we eliminate the cross term using rotation.
在解析几何中，一般二次方程 ax² + 2bxy + cy² + ... = 0 表示一条圆锥曲线。二次部分 ax² + 2bxy + cy² 在通过旋转消去交叉项后，决定了圆锥曲线的类型（椭圆、抛物线、双曲线）。
it is positive definite, the conic is an ellipse; if indefinite, a hyperbola; if semi-definite and degenerate, a parabola. The signs of eigenvalues correspond to the signs of the coefficients in the rotated form.
二次型 Q(x,y) = ax² + 2bxy + cy² 的定号性告诉我们：若它是正定的，圆锥曲线是椭圆；若不定，则是双曲线；若半定且退化，则是抛物线。特征值的符号与旋转后表达式中各项系数的符号相对应。
In optimisation, for a function f whose Hessian matrix is H, the second-order Taylor term is a quadratic form (1/2) ΔxT H Δx. Positive definiteness of H at a stationary point confirms a local minimum, negative definiteness a local maximum, and indefinite a saddle point.
在优化问题中，对于函数 f 其海森矩阵为 H，二阶泰勒项即为一个二次型 (1/2) ΔxT H Δx。在驻点处 H 的正定性确认局部极小值，负定性确认局部极大值，不定性则意味着鞍点。
10. Summary and Exam Tips | 总结与考试要点
When tackling quadratic form questions, always start by writing down the symmetric matrix A correctly – check the off-diagonal entries by halving cross-term coefficients. Identify the rank and use either eigenvalue or principal minor criteria to classify definiteness.
解二次型题目时，始终先从正确写出对称矩阵 A 入手——核对非对角线元素是否等于交叉项系数的一半。确定秩，并运用特征值或主子式判据分类定号性。
For simplification, completing the square works well for 2 or 3 variables and often appears in Paper 1 without calculators. For larger systems or when eigenvalues are requested, orthogonal diagonalisation is the systematic route. Remember that the orthogonal matrix P contains normalised eigenvectors.
对于化简，配方法在 2 或 3 个变量时非常有效，且常出现在不允许使用计算器的 Paper 1 中。对于更大系统或需要求特征值时，正交对角化是系统的方法。记住，正交矩阵 P 包含的是标准化的特征向量。
Pay close attention to the wording: 'positive definite', 'negative semi-definite', etc., have precise definitions. Sketching the transformed sum of squares helps visualise the geometric meaning and avoid sign errors.
仔细审题：“正定”、“半负定”等术语都有精确定义。画出变换后的平方和示意图，有助于直观理解几何意义并避免符号错误。
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