A Geometric Interpretation of the Principal Invariants of a Real

Symmetric Matrix

Roberto B. Salgado

June 15, 1995

Abstract

We present a geometric interpretation of the principal invariants of a real symmetric n-by-n matrix. We will show that the k^{th} principal invariant (where 1 ≤ k ≤ n) is a measure of the bounding k-volume of an n-dimensional rectangular parallelopiped.

1 Introduction

The k^{th} principal invariant I_{k} = A^{[a}_{a} ⋯ A^{k]}_{k} is the coefficient of the λ^{n−k+1} term in the characteristic polynomial of A^{a}_{b}. It is equal to the k^{th} elementary symmetric function of the eigenvalues of A^{a}_{b}.

Let Q(k, n) denote the number of k-cubes in an n-cube.

We need the number of k-cubes at each vertex.

We need the number if edges at each vertex: this number is n.

To form a k-cube, one chooses k distinct edges from those n edges: the number of ways to do this is \frac{n!}{k! (n−k)!} .

The number of vertices in an n cube is 2^{n}.

Hence,

Q(k, n) = \frac{(2^{n}) (\frac{n!}{k! (n−k)!})}{2^{k}} ,

since each k-cube is counted 2^{k} times.

Hence, the k^{th}-invariant

I_{k} = \frac{1}{2^{n−k}} Q̃(k, n),

where Q̃(k, n) is the k-volume-weighted number of k-cubes in this n-prism.

References

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Department of Physics Syracuse University Syracuse NY 13244-1130

salgado@suhep.phy.syr.edu