Question

Find the getschgorin bounds on the eigenvalurd of the matrix

Answer 1

Because that is a stochastic matrix you just need to fine the fixed probability vector or steady state vector.

(t1,t2)⋅(0.90.20.10.8)=(t1,t2)

Notice I have transposed A for convenience and this yields the simultaneous set of equations

.9⋅t1+.2⋅t2=t1

.1⋅t1+.8⋅t2=t2

t1+t2=1

this is easily solved to get

t1=23

t2=13

Because this is a regular Markov chain all the rows of A∞ are equal to (t1,t2), so

A∞=(23231313)

Transpose A back:

A∞=(23132313)

Hope this helps you ☺️☺️✌️✌️