Lecture Notes for 416, Inner Products and Spectral Theorems

Real inner product. Let V be a vector space over R. A (real) inner product is a function 〈−,−〉 : V × V → R such that • 〈x, y〉 = 〈y, x〉 for all x, y ∈ V, • 〈c1x1 + c2x2, y〉 = c1〈x1, y〉+ c2〈x2, y〉 for all x1, x2, y ∈ V, c1, c2 ∈ R, • 〈x, x〉 ≥ 0 with 〈x, x〉 = 0 iff x = 0. That is, the pairing is symmetric, linear in the first variable (and therefore bilinear, by symmetry), and positive definite. Example (Standard dot product). Given a column vector v ∈ Rn×1, let vt ∈ R1×n be the transpose. Then define 〈x, y〉 = yx. Check that this is just the usual dot product, so that 〈x, y〉 = ∑ xiyi. Example. Let P ∈ Rn×n be an invertible matrix. Define 〈x, y〉P = yP Px. This is an inner product, usually different from the dot product. Fact: every inner product on Rn×1 is of this form for some P . (See discussion of isometries below.)