Construct the transition matrix describing the pond this bullfrog will be in$,$ and save it as a NumPy array called trans$\_$matrix. When ordering your variables to create the matrix, order the ponds as A$,$ B$,$ C$,$ then D$.$

Interactive

Find the steady state probability vector for this transition matrix, and determine what percentage of the bullfrog's time will be spent in pond C in the long run $($regardless of which pond he starts in$).$ Recall that finding the steady state probability vector is the same thing as finding the eigenvector corresponding to the dominant eigenvalue $1.$ You don't need to submit the answer you find for this steady state vector.

Hint

Use a function you defined earlier.

Previous functions used:

A$=$np$.$array$([[1,1],[2,0]])$

def evect$\_$approx$1($x$\_0,$k$)$:

x$\_$vect$=0$

for j in range$(1,$k$+1)$:

x$\_$vect$=$np$.$dot$($A$,$x$\_0)$

x$\_0=$x$\_$vect

return x$\_$vect

# This function approximates the dominant eigenvalue of our matrix A$.$

def eval$\_$approx$1($x$\_0,$k$)$:

A$=$np$.$array$([[1,1],[2,0]])$

lambda$\_1=0$

for j in range$(1,$k$+1)$:

x$\_$vect$=$np$.$dot$($A$,$x$\_0)$

x$\_0=$x$\_$vect

lambda$\_1=($np$.$dot$($A$,$x$\_$vect$)/$x$\_0)[0]$

return lambda$\_1$

# This function approximates the dominant eigenvalue and eigenvector of our matrix A using the normalized iterative process.

def norm$\_$evect$\_$approx$1($x$\_0,$ k$)$:

A $=$ np$.$array$([[1,1],[2,0]])$

for j in range$($k$)$:

w$\_$j $=$ A @ x$\_0$

x$\_$j $=$ x$\_0$

x$\_0=$ w$\_$j $/$ np$.$linalg.norm$($w$\_$j$)$

val $=$ w$\_$j$[0]/$ x$\_$j$[0]$

return x$\_0,$ val

# This function approximates the dominant eigenvalue and eigenvector of our matrix A using the normalized iterative process.

def norm$\_$evect$\_$approx$1($x$\_0,$ k$)$:

A $=$ np$.$array$([[1,1],[2,0]])$

for j in range$($k$)$:

w$\_$j $=$ A @ x$\_0$

x$\_$j $=$ x$\_0$

x$\_0=$ w$\_$j $/$ np$.$linalg.norm$($w$\_$j$)$

val $=$ w$\_$j$[0]/$ x$\_$j$[0]$

return x$\_0,$ val

# This function approximates the dominant eigenvalue and eigenvector of an arbitrary matrix using the Rayleigh quotiend as described in Problem $5.$

def ray$\_$quotient$($M$,$ x$\_0,$ k$)$:

x$\_$vect, $\_=$ norm$\_$approx$\_$gen$($M$,$ x$\_0,$ k$)$

rayleigh$\_$quotient $=$ np$.$dot$($np$.$dot$($M$,$ x$\_$vect$),$ x$\_$vect$)/$ np$.$dot$($x$\_$vect, x$\_$vect$)$

return rayleigh$\_$quotient

def subscriber$\_$vals$($x$\_0,$ k$)$:

P $=$ np$.$array$([[0\mathrm{.}7,0\mathrm{.}2],$

$[0\mathrm{.}3,0\mathrm{.}8]])$

x $=$ x$\_0$

for $\_$ in range$($k$)$:

x $=$ np$.$dot$($P$,$ x$)$

return x