Question on eigenvector calculation

[mikeraj]

I have a question regarding one example of eigenvector calculation.

 

In Equation (1) of the attached example, should the first column of matrix A be written as 0.8x₁ + 0.2x₂ , rather than x₁ + 0.2x₂ ?

 

Thanks in advance for your inputs.

example.pdf

Edited October 31, 2016 by mikeraj

[Country Boy]

To write the first column of A, (.8, .2), as a linear combination of x1 and x2 we need to find numbers a and b such that (.8, .2)= a(.6, .4)+ b(1, -1)= (.6a+ b. .4a- b). That gives the two equation .6a+ b= .8 and .4a- b= .2. Adding the two equations eliminates b: (.6+ .4)a= a= .8+ .2= 1.0. With a= 1. .6a+ b= .6+ b= .8 so b= 0.2. It is 0.8x+ 0.2x2.

Are you clear on what "x1" and "x2" are? Equation (1) writes (0.8, 0.2) as a linear combination of x1= (0.6, 0.4) and x2= (1, -1). That means we need to find numbers, a and b, such that a(0.6, 0.4)+ b(1, -1)= (0.6a+ b. 0.4a- b)= (0.8, 0.2). That gives the two equations 0.6a+ b= 0.8 and 0.4a- b= 0.2. Adding the two equations eliminates b: (0.6+ 0.4)a= a= 0.8+ 0.2= 1.0. Since a= 1, 0.6a+ b= 0.6+ b= 0.8 so b= 0.8- 0.6= 0.2. (0.8, 0.2)= 0.8(0.6, 0.4)+ 0.2(1, -1) is correct.

[mikeraj]

Hi HallsofIvy, I am clear now. Many thanks again!