Linear Algebra and Its Applications · Problem 6.4.2

Computations with Matrices — Gilbert Strang

Mathematics · minimum-principles-completing-square

Problem

Complete the square in

P = \frac{1}{2}x^TAx - x^Tb = \frac{1}{2}(x - A^{-1}b)^TA(x - A^{-1}b) + \text{constant}

. This constant equals

P_{\min}

because the term before it is never negative. (Why?)