1. Find Power Method for finding dominant eigenvalue ...
`[[2,3],[5,4]]`
`x_0` = 1,1

Solution:

`A=`

2 3 5 4

`x_0=`

1 1

`1^(st)` iteration :

Multiply the matrix by the vector

`A x_0 =`

2 3 5 4

1 1

`=`

5 9

Normalize the resulting vector
To normalize, divide each element of vector by its largest absolute value, which is `9`

`x_1=`

`1/9`

5 9

`=`

0.5556 1

`2^(nd)` iteration :

Repeat the multiplication

`A x_1 =`

2 3 5 4

0.5556 1

`=`

4.1111 6.7778

Normalize again
The largest absolute value is `6.7778`

`x_2=`

`1/6.7778`

4.1111 6.7778

`=`

0.6066 1

`3^(rd)` iteration :

Repeat the multiplication

`A x_2 =`

2 3 5 4

0.6066 1

`=`

4.2131 7.0328

Normalize again
The largest absolute value is `7.0328`

`x_3=`

`1/7.0328`

4.2131 7.0328

`=`

0.5991 1

`4^(th)` iteration :

Repeat the multiplication

`A x_3 =`

2 3 5 4

0.5991 1

`=`

4.1981 6.9953

Normalize again
The largest absolute value is `6.9953`

`x_4=`

`1/6.9953`

4.1981 6.9953

`=`

0.6001 1

`5^(th)` iteration :

Repeat the multiplication

`A x_4 =`

2 3 5 4

0.6001 1

`=`

4.2003 7.0007

Normalize again
The largest absolute value is `7.0007`

`x_5=`

`1/7.0007`

4.2003 7.0007

`=`

0.6 1

`6^(th)` iteration :

Repeat the multiplication

`A x_5 =`

2 3 5 4

0.6 1

`=`

4.2 6.9999

Normalize again
The largest absolute value is `6.9999`

`x_6=`

`1/6.9999`

4.2 6.9999

`=`

0.6 1

`:.` The dominant eigenvalue `lamda=6.9999~=7`

and the dominant eigenvector is :

`=`

0.6 1