Question

find the spectral radius of A. Is this a convergent matrix? Justify your answer. Find the limit x=lim x^(k) of vector iteration

Answer 1

The solution to this question can be defined as follows:

Explanation:

Please find the complete question in the attached file.

`A = \left[\begin{array}{ccc} (3)/(4)& (1)/(4)& (1)/(2)\\ 0 & (1)/(2)& 0\\ -(1)/(4)& -(1)/(4) & 0\end{array}\right]`

now for given values:

`\left[\begin{array}{ccc} (3)/(4) - \lambda & (1)/(4)& (1)/(2)\\ 0 & (1)/(2) - \lambda & 0\\ -(1)/(4)& -(1)/(4) & 0 -\lambda \end{array}\right]=0 \\\\`

`\to ((3)/(4) - \lambda ) [-\lambda ((1)/(2) - \lambda ) -0] - 0 - (1)/(4)[0- (1)/(2) ((1)/(2) - \lambda )] =0 \\\\\to ((3)/(4) - \lambda ) [((\lambda)/(2) + \lambda^2 )] - (1)/(4)[(\lambda)/(2) - (1)/(4)] =0 \\\\\to ((3)/(8)\lambda + (3)/(4) \lambda^2 - (\lambda^2)/(2) - \lambda^3 - (\lambda)/(8) + (1)/(16)=0 \\\\\to (\lambda - (1)/(2)) (\lambda -(1)/(4)) (\lambda - (1)/(2)) =0\\\\`

`\to \lambda_1=\lambda_2 =(1)/(2)\\\\\to \lambda_3 = (1)/(4) \\\\\to A = max \\\\`

`= max{(1)/(2), (1)/(2), (1)/(4)}\\\\= (1)/(2)\\\\(A) =(1)/(2)`

In point b:

Its

spectral radius is less than 1 hence matrix is convergent.

In point c:

`\to c^((k+1)) = A x^(k)+C \\\\\to x(0) = \left(\begin{array}{c}3&1&2\end{array}\right) , c = \left(\begin{array}{c}2&2&4\end{array}\right)\\\\ \to x^((k+1)) = \left[\begin{array}{ccc} (3)/(4)& (1)/(4)& (1)/(2)\\ 0 & (1)/(2)& 0\\ -(1)/(4)& -(1)/(4) & 0\end{array}\right] x^k + \left[\begin{array}{c}2&2&4\end{array}\right] \\\\`

after solving the value the answer is

:

`\lim_(k \to \infty) x^k=o = \left[\begin{array}{c}0&0&0\end{array}\right]`