Description
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:
False Proof: Ax = λx gives xTAx = λxTx so λ = x
x
T
T
A
x
x is real.
There is a hidden assumption in this proof which is not justified. Find the
flaw by testing each step on the 90 ◦ rotation matrix:
0 −1
1 0
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.
1
18.06SC Linear Algebra
