[Solved] 18.06 Unit 2 Exercise 11- Markov matrices; Fourier series

Problem 24.1: (6.4 #7. Introduction to Linear Algebra: Strang)

1. Find a symmetric matrix that has a negative eigenvalue.
2. How do you know it must have a negative pivot?
3. 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

QΛQ^T?

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 .

2