Exercises on Markov matrices; Fourier series
Problem 24.1: (6.4 #7. Introduction to Linear Algebra: Strang)
a) Find a symmetric matrix that has a negative eigenvalue.
b) How do you know it must have a negative pivot?
c) How do you know it can’t have two negative eigenvalues?
Problem 24.2: (6.4 #23.) Which of these classes of matrices do A and B belong to: invertible, orthogonal, projection, permutation, diagonalizable,
Markov?
A B .
Which of these factorizations are possible for A and B: LU, QR, SΛS−1, or
QΛQT?
1
Problem 24.3: (8.3 #11.) Complete A to a Markov matrix and find the steady state eigenvector. When A is a symmetric Markov matrix, why is x1 = (1, . . . , 1) its steady state?
⎡
A = ⎣ .1 .
