matrix power and summation

Hi, I was wondering how to calculate matrix with power and Summation in matlab, for example, for matrix A, how to calculate A+A^2+A^3....A^n

Thanks

Answer 1

[Steven Lord]

Consider using the polyvalm function.

Answer 2

[Walter Roberson]

total = A;
for K = 2 : n
  total = total + A^K;
end

Answer 3

[John D'Errico]

Why do people never use the basic methods available? (Only kidding, as I suppose this is not obvious that what I show is valid for a matrix A. It is.) In this case, this is just an n-term geometric series, with a "ratio" between terms of A.

https://en.wikipedia.org/wiki/Geometric_series

Yes, I know that everyone thinks about this as appropriate for SCALARS. In fact however, as long as the matrix A does not have an eigenvalue of 1, we have a simple formula that is completely valid. First, the loops form to convince you that it is valid.

A = rand(3);
n = 20;
B = A;
for i = 2:n
  B = B + A^i;
end

B
B =
        336.893406676369          454.133952892778          295.131937851577
        193.036906910648           259.86866195271          168.859953528597
        311.848524182241          419.850392065738          272.579343092041

So that is what you get from the loop. How about a simple algebraic expression using matrices?

C = A*(eye(size(A)) - A^n)*pinv(eye(size(A)) - A)
C =
        336.893406676369          454.133952892778          295.131937851577
        193.036906910648           259.86866195271          168.859953528597
        311.848524182241          419.850392065738          272.579343092041

This should be more efficient than a loop, certainly is n is large.

The expression I wrote is simply the standard form for the sum of a finite geometric series, adapted for matrices. The reason it fails is if A has any eigenvalues exactly equal to 1, then the pinv computation will be inappropriate here. Otherwise, this result will be exact.

[Walter Roberson]

Cool approach for the infinite series!

Answer 4

[Leonidas Savvides]

exist any special function in MatLab find power of a Matrix? like above

Also any function in MatLab in Discrete Math [Relations] find connectivity relation R* that is [M]R∗ = M ∨ M^[2] ∨ M^[3] ∨ · · · ∨ M^[n] ...?

If no Exist then ... how calculate matrix=M=M1 ∧ M2 or M=M1 ∨ M2...?

[Walter Roberson]

Leonidas Savvides: Are you looking for sum of X^k for matrix X, k = 1 to n? Or are you looking for sum of (X^k)/factorial(k) for matrix X, k = 0 to infinity? If you happen to be looking for that, then expm() calculates that.

Is your ∨ symbol intend to indicate element-by-element "or", and is your ∧ symbol intended to indicate element-by-element "and" ? Would those be binary matrices? Is the first of those intended to be "the set of places that can be reached from each starting node and taking at most n steps" ? The interpretation of the other does not spring to my mind at the moment.

[John D'Errico]

Moved an answer intended as a comment by Leonids to a comment. Please stop adding answers every time you want to make a comment. Use the blue comment link instead.

"Are you looking for sum of X^k for matrix X, k = 1 to n? YES , but Not the other Not factorial involved

yes, all binary matrices only 1 and 0 elements...

Is the first of those intended to be "the set of places that can be reached from each starting node and taking at most n steps" ? YES

Is your ∨ symbol intend to indicate element-by-element "or", and is your ∧ symbol intended to indicate element-by-element "and" ? YES"

Answer 5

[Jan]

The power operation is expensive. But you can avoid it:

S  = A;
AA = A;
for K = 2 : n
  AA = AA * A;
  S  = S + AA;
end