Bilinear Forms and Tensor Products

University:
The University Of Western Ontario
Course:
MATH 1620B | Discrete Mathematic
Academic year:

2021
Views:

159

Pages:

3
Author:

Kyuwon J.

Study Notes: Bilinear Forms and Tensor Products (Part 1) Introduction to Bilinear Forms and Quadratic Forms Key Definitions Bilinear Forms: • A bilinear form f : V × V → R satisfies linearity in each argument: f (ax + by, z) = af (x, z) + bf (y, z), f (x, ay + bz) = af (x, y) + bf (x, z). • Symmetry: A bilinear form f (x, y) is symmetric if f (x, y) = f (y, x) for all x, y ∈ V . • Alternating: A bilinear form f (x, x) = 0 for all x ∈ V is alternating. • Nondegeneracy: A bilinear form f is nondegenerate if f (x, y) = 0 for all y ∈ V implies x = 0. Quadratic Forms: • A quadratic form is defined as Q(x) = f (x, x), where f is a bilinear form. • Quadratic forms can often be represented in matrix form and diagonalized using eigenvalues. 1 Bilinear Forms: Examples and Solutions Example 1: Symmetry of a Bilinear Form • Let f (x, y) = x1 y1 +x2 y2 . Prove symmetry and compute f ((1, 2), (3, 4)). Solution: • Symmetry: f (x, y) = x1 y1 + x2 y2 = y1 x1 + y2 x2 = f (y, x). • Computation: f ((1, 2), (3, 4)) = 1 · 3 + 2 · 4 = 3 + 8 = 11. Example 2: Quadratic Form from a Bilinear Form • Let g(x, y) = x1 y1 + 2x1 y2 + x2 y2 . Find the quadratic form f (x) = g(x, x). Solution: • Substitute y = x in g(x, y): f (x) = g(x, x) = x21 + 2x1 x2 + x22 . Example 3: Alternating and Nondegenerate Bilinear Form • Show that f (x, y) = x1 y2 − x2 y1 is alternating and nondegenerate. Solution: • Alternating: f (x, x) = x1 x2 − x2 x1 = 0. • Nondegeneracy: For x ̸= 0, there exists y such that f (x, y) ̸= 0. Choose y = (x2 , −x1 ): f (x, y) = x1 (−x1 ) − x2 x2 = −x21 − x22 ̸= 0. 2 Matrix Representation of Bilinear Forms Symmetric and Alternating Parts of a Matrix: • For a matrix A, the symmetric part As and alternating part Aa are given by: 1 1 As = (A + A⊤ ), Aa = (A − A⊤ ). 2 2 Example: 1 2 • Let A = . 3 4 • Compute the symmetric 1 1 As = 2 3 1 1 Aa = 2 3 and alternating parts: 1 1 3 2 1 2.5 + = . 4 2.5 4 2 2 4 1 1 3 2 0 −0.5 − = . 4 0.5 0 2 2 4 Visualizing Bilinear Forms Diagram 1: Symmetry in Bilinear Forms x1 x2 f (x1 , y1 ) y1 y2 f (x2 , y2 ) Diagram 2: Symmetric and Alternating Components 1 2 1 2.5 0 −0.5 = + 3 4 2.5 4 0.5 0 | {z } | {z } As Aa Applications and Summary 3