A stochastic matrix shows us that
the sum all elements on each column (or each row) is equal one or we can say
that each column (or each row) of a stochastic matrix is just a probability
vector.
Theorem (1):
The sequence of operations a stochastic matrix on any vector will be finally
obtained the constant vector. (The elements of vector are the members of Real Number)
V1 = V0 * M^1 ,
V2 = V1*M^2, V3 = V2*M^3, …. Vn+1 = V n*
M^k
Where vector
Vn+1 always will be the constant.
A special state is when the vector is a probability
vector which is named Markov Chain or Markov Process
Example (1):
The sample of our vector (1*4) is:
V0 =
Suppose we have below stochastic
matrix M (4*4):
M =
The results of the sequences (V0 * M) are as
follows:
As we can see, the vectors of V5 and
V6 are approximately the same.
Example (2):
The sample of our vector (1*4) is:
V0 =
The results of the sequences are as
follows:
As we can see, the vectors of V5 and
V6 are approximately the same.
An example for
Markov Chain:
The sample of our probability vector
(1*4) is:
V0 =
I consider above stochastic matrix M
(4*4).
Then, we will have below results:
As we can see, the vectors of V5 and
V6 are approximately the same.
Is below conjecture is a theorem?
Conjecture (1):
The sequence of operations for transpose of a stochastic matrix (n*n) with non - zero elements on any vector
will be finally obtained a vector with the same elements. (The elements of vector are the members of Real Number)
Consider above stochastic matrix M
(4*4):
M =
The transpose of matrix M is:
MT =
The sample of vector is:
V0 =
The results of the sequences are as
follows:
As we can see, the elements of vector V7 are the same.
Example (2):
The sample of vector is:
V0 =
The results of the sequences are as
follows:
As we can see, the elements of vector V7 are the same.
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