PROOFS FOR FINDING THE MAXIMA OF TWO QUADRATIC FORMS UNDER SOME CONSTRAINTS

Abstract

A quadratic form on ${\mathrm{ℝ}}^n$ is a function $Q$ defined on ${\mathrm{ℝ}}^n$ whose value at a vector $x$ in ${\mathrm{ℝ}}^n$ is an expression of the form $Q\left(x\right)=x^{T}Ax$, where $A$ is $n\times n$ symmetric matrix. In this note we discuss how to prove two theorems, Theorem A and Theorem B, for finding the maxima of two quadratic forms under some constraints. Of course, the proofs of these theorems are well known. But in this note we give an improved proof of Theorem A and a simpler proof of Theorem B.