Borevich-Shafarevich《数论》中二次型等价性及相关内容技术问询

Let's dive into quadratic form equivalence and the change of variables you're exploring, using the foundational setup from Borevich-Shafarevich's Number Theory that you referenced.

Background Recap (from Borevich-Shafarevich) #

任意二次型$f$可表示为$$\sum_{i,j=1}^{n} a_{i,j}x_i x_j,$$其中$a_{i,j}=a_{j,i}$。对称矩阵$A=(a_{ij})$称为二次型$f$的矩阵。若用$X$表示变量$x_1,\dots,x_n$构成的列向量，则二次型可写为$$f = X^tAX.$$

What Happens When We Switch to New Variables $y_1,\dots,y_n$? #

Suppose we have an invertible linear transformation mapping the old variable column vector $X$ to the new column vector $Y$:
$$X = PY$$
Here, $P$ is an $n \times n$ invertible matrix—this is critical because we need to be able to reverse the transformation, preserving the full dimensionality of the variable space.

Substitute this into the original quadratic form:
$$f = (PY)^tA(PY) = Y^t(PtAP)Y$$
The matrix corresponding to the transformed quadratic form is $B = P^tAP$, which remains symmetric (since $B^t = (P^tAP)t = P^tAtP = P^tAP = B$, given $A$ is symmetric).

Defining Quadratic Form Equivalence #

Two quadratic forms $f$ (with matrix $A$) and $g$ (with matrix $B$) are equivalent if and only if there exists an invertible matrix $P$ such that $B = P^tAP$.

Put simply, equivalent quadratic forms are just the same quadratic function viewed through different coordinate systems—they represent the same quadratic structure on the vector space, just expressed relative to a different basis (where variables are the coordinates of vectors in that basis).

Key Properties of Equivalent Quadratic Forms #

• Rank Preservation: Equivalent quadratic forms have identical rank. Invertible transformations don't change a matrix's rank, so $\text{rank}(P^tAP) = \text{rank}(A)$.
• Signature Invariance (over $\mathbb{R}$): Over the real numbers, equivalent quadratic forms share the same inertia (count of positive, negative, and zero eigenvalues, per Sylvester's Law of Inertia). This is the core of classifying real quadratic forms.
• Determinant Relationship: The determinant of the transformed matrix relates to the original via $\det(B) = (\det(P))^2\det(A)$. Since $(\det(P))^2$ is always non-negative (and positive, because $P$ is invertible), the sign of the determinant is preserved over $\mathbb{R}$, and $\det(B)$ is zero if and only if $\det(A)$ is zero.

General Recommendations for Working with Equivalent Quadratic Forms #

• Stick to Invertible Transformations: Only use invertible matrices $P$ for variable changes. Non-invertible transformations collapse the variable space, resulting in a restricted version of the original quadratic form—not an equivalent one.
• Reduce to Canonical Forms: For any quadratic form, you can always find an invertible transformation to convert it to a canonical (standard) form:
  • Over $\mathbb{R}$: This is $\sum_{i=1}^p y_i^2 - \sum_{i=p+1}^{p+q} y_i^2$, where $p$ is the positive inertia index and $q$ is the negative inertia index.
  • Over $\mathbb{C}$: This simplifies to $\sum_{i=1}^r y_i^2$, where $r$ is the rank of the quadratic form.
• Account for the Base Field: Equivalence depends heavily on the field you're working over. For example, $2x_1^2 + 2x_2^2$ is equivalent to $x_1^2 + x_2^2$ over $\mathbb{R}$ (using $P = \frac{1}{\sqrt{2}}I$), but not over $\mathbb{Q}$ (since $\sqrt{2}$ isn't rational).
• Leverage Matrix Diagonalization: Use techniques like completing the square or orthogonal diagonalization (for real quadratic forms) to find the appropriate transformation matrix $P$. Orthogonal transformations are especially useful over $\mathbb{R}$ because they preserve inner products, corresponding to rigid geometric transformations (rotations/reflections).

内容的提问来源于stack exchange，提问作者Davide Gallo