# Perron formula for matrix powers

• bpet
In summary, a formula for determining the elements of matrix P^n using the Perron formula has been found in an old probability textbook. This formula works for any matrix and is not related to the Perron formula of number theory. It involves finding the Laplace transform of e^{tP} and extracting the nth term of the Taylor series. It can also be derived by finding the residue of \lambda^n(\lambda I-P)^{-1} at \lambda=\infty via Laurent series and partial fraction expansion. The formula was also mentioned in Sveshnikov's Problems in Probability and may be discussed in other texts such as Gantmakher's Theory of Matrices or Horn & Johnson's Matrix Analysis.
bpet
The following interesting result popped up in an old probability textbook (without proof or citations) and I'm curious to know how it can be derived.

The elements $$p_{ij}^{(n)}$$ of matrix $$P^n$$ (size m) are also determined by the Perron formula

$$p_{ij}^{(n)} = \sum_{s=1}^r \frac{1}{(v_s-1)!}\left\{\frac{d^{v_s-1}}{d\lambda^{v_s-1}}\left[\frac{\lambda^nA_{ji}(\lambda)(\lambda-\lambda_s)^{v_s}}{|\lambda I_m-P|}\right]\right\}_{\lambda=\lambda_s}$$

where $$r$$ is the number of distinct eigenvalues, $$\lambda_s$$ are the distinct eigenvalues with multiplicity $$v_s$$ (so $$v_1+\ldots+v_s=m$$) and $$A_{ji}(\lambda)$$ is the cofactor of the elements $$\lambda\delta_{ji}-p_{ji}$$ in the determinant $$|\lambda I_m-P|$$ and $$I_m$$ is the identity matrix of the same size as P.

It seems to work for any matrix (not just Markov transition probabilities) and as far as I can tell it's not related to the Perron formula of number theory.

The term $$A_{ji}(\lambda)/|\lambda I_m-P|$$ would hint that $$(\lambda I_m-P)^{-1}$$ is involved, so I suspect it's done by finding the Laplace transform of $$e^{tP}$$ and somehow extracting the nth term of the Taylor series. Is this on the right track and if so, how would it be done? In particular, how do they turn the expression into a sum of derivatives at the eigenvalues?

Have you managed to derive the formula yet? If I had time I'd love to have a go...

Anyway, I did a little searching and found
http://crypto.mat.sbg.ac.at/~ste/diss/node12.html
There he cites pg 16 of
V. Romanovsky, Discrete Markov Chains.
but I couldn't get a copy of it.

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Simon_Tyler said:
Have you managed to derive the formula yet? If I had time I'd love to have a go...

Anyway, I did a little searching and found
http://crypto.mat.sbg.ac.at/~ste/diss/node12.html
There he cites pg 16 of
V. Romanovsky, Discrete Markov Chains.
but I couldn't get a copy of it.

Thanks, me neither. I haven't checked the details but think it can be done by finding the residue of $$\lambda^n(\lambda I-P)^{-1}$$ at $$\lambda=\infty$$ via Laurent series and partial fraction expansion.

The formula was in Sveshnikov's Problems in Probability (as a "basic formula", not an actual problem). The result might be discussed in Gantmakher's Theory of Matrices or Horn & Johnson's Matrix Analysis, possibly even for more general matrix functions, though I don't have copies of these to check.

Last edited by a moderator:

## 1) What is the Perron formula for matrix powers?

The Perron formula for matrix powers is a mathematical formula that allows for the computation of the powers of a square matrix by using its eigenvalues and eigenvectors. It states that the kth power of a matrix A is equal to the sum of the products of its eigenvalues raised to the power of k and their corresponding eigenvectors.

## 2) How is the Perron formula used in practical applications?

The Perron formula is commonly used in the fields of physics, engineering, and economics to model and analyze systems with a repetitive structure, such as electrical circuits or population dynamics. It is also used in computer science and data analysis for tasks such as image processing and pattern recognition.

## 3) Can the Perron formula be used for non-square matrices?

No, the Perron formula is only applicable to square matrices, meaning matrices with an equal number of rows and columns. This is because it relies on the concept of eigenvalues and eigenvectors, which are only defined for square matrices.

## 4) How does the Perron formula relate to the Perron-Frobenius theorem?

The Perron formula is closely related to the Perron-Frobenius theorem, which states that for a non-negative square matrix, the largest eigenvalue is real and positive, and its corresponding eigenvector has strictly positive components. The Perron formula is derived from this theorem and allows for the computation of the powers of a matrix using its largest eigenvalue and eigenvector.

## 5) Are there any limitations or assumptions to consider when using the Perron formula?

Yes, there are a few limitations and assumptions to keep in mind when using the Perron formula. First, it only applies to square matrices with real eigenvalues. Additionally, it assumes that the matrix is non-negative and has a dominant eigenvalue (largest in magnitude). It also requires the matrix to be diagonalizable, meaning it can be written as a product of its eigenvalues and eigenvectors.

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