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分类： Matrix identity

  • 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 n\le3). Is it classical ?

    I am given a real symmetric matrix
    S:=\int Y(t)Y(t)^Td\mu(t),
    where \mu is a probability and Y(t):\Omega\rightarrow{\mathbb R}^n.

    Let \sigma_k(S) be the elementary symmetric polynomial in the eigenvalues of S. For instance, \sigma_1(S) is the trace and \sigma_n(S) the determinant. The following formula gives \sigma_k(S) in terms of the Gram matrix G_k(s_1,\ldots,s_k) whose entries are the scalar products Y(s_i)\cdot Y(s_j).

    \sigma_k(S)=\frac1{k!}\int^{\otimes k}\det G_k(s_1,\ldots,s_k)\,d\mu(s_1)\cdots d\mu(s_k).

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

    The case k=n is a consequence of the identity

    \int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)

    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。

    一个 determinantal formula：Gram identity、Andreief identity 与对称多项式
    Gram identity 和 Andreief identity 把行列式、体积平方和对称多项式联系起来。

    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。