一个 determinantal formula：Gram identity、Andreief identity 与对称多项式

旧博客原文

原题：A determinantal formula

I see a similar formula I wish to be true and merely have a proof in mind occur as a MO’s problem:

In my research, I encounter the following formula which I believe is correct (checked for ). Is it classical ?

I am given a real symmetric matrix

where is a probability and .

Let be the elementary symmetric polynomial in the eigenvalues of . For instance, is the trace and the determinant. The following formula gives in terms of the Gram matrix whose entries are the scalar products .

Remark that is positive semi-definite. The integrand is non-negative, as well as . The integrand vanishes identically iff takes values in a subspace of dimension , which is the condition under which vanishes. It follows that, if the formula above failed, it would be because of an inequality between strictly positive numbers.

The case is a consequence of the identity

which I have seen under the names “Andreief identity” and also “Gram identity”. The proof is elementary using the Leibniz formula for the determinant.

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

很多行列式公式可以理解为 Gram identity 或 Andreief identity 的变体。它们把矩阵的对称多项式、向量组的体积和积分中的 determinant 联系起来。

1. Gram matrix

给定向量 $v_1,\ldots,v_m$，Gram matrix 为

$$G_{ij}=\langle v_i,v_j\rangle.$$

$\det G$ 等于这些向量张成平行体体积的平方。因此 $G$ 半正定，且 $\det G=0$ 当且仅当向量线性相关。

2. Andreief identity

Andreief identity 的典型形式是

$$\int \det(f_i(x_j))\det(g_i(x_j))\prod_j d\mu(x_j)
=n!\det\left(\int f_i(x)g_j(x)d\mu(x)\right).$$

它是积分版的 Cauchy-Binet 公式。

3. 对称多项式

若 $A$ 是半正定矩阵，特征值为 $\lambda_1,\ldots,\lambda_n$，第 $k$ 个 elementary symmetric polynomial 是

$$e_k(A)=\sum_{i_1<\cdots

它也等于所有 $k\times k$ 主子式之和。

4. 几何解释

$e_k(A)$ 可以看成 $A$ 在所有 $k$ 维方向上的体积膨胀总和。若 $A$ 来自随机向量的 Gram matrix，那么对应公式会把 $e_k(A)$ 写成某种 determinant 的积分平均。

5. 证明方式

这类公式通常从 Leibniz determinant expansion 出发，交换求和和积分，再识别为 Cauchy-Binet 或 Andreief identity。核心不是计算技巧，而是“行列式等于体积平方”的几何含义。