Exercises on symmetric matrices and positive definiteness
Problem 25.1: (6.4 #10. Introduction to Linear Algebra: Strang) Here is a quick “proof” that the eigenvalues of all real matrices are real:
TAx = λxTx so λ = xxTTAxx is real. False Proof: Ax = λx gives x
There is a hidden assumption in this proof which is not justified. Find the flaw by testing each step on the 90 ◦ rotation matrix:
with λ = i and x = (i,1).
Problem 25.2: (6.5 #32.) A group of nonsingular matrices includes AB and
A−1 if it includes A and B. “Products and inverses stay in the group.” Which of these are groups?
a) Positive definite symmetric matrices A.
b) Orthogonal matrices Q.
c) All exponentials etA of a fixed matrix A.
d) Matrices D with determinant 1.
