Finding powers of a square matrix

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Discussion Overview

The discussion focuses on methods for calculating the powers of a square matrix, specifically exploring the Putzer Algorithm, the Cayley-Hamilton Theorem, and the Jordan form. Participants are considering various approaches and their efficiency in evaluating A^n for integer n greater than 1.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant introduces the Putzer Algorithm as a method for calculating higher powers of a square matrix.
  • Another suggests that diagonalization can be used to exponentiate the matrix, implying it is a straightforward method.
  • A different participant counters that not all matrices can be diagonalized, referencing the Jordan form as a necessary consideration for non-diagonalizable cases.
  • Another approach mentioned is using QR decomposition to obtain a good approximation for the matrix powers.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of diagonalization, with some supporting it as a method while others highlight its limitations. The discussion remains unresolved regarding the most efficient method for calculating matrix powers.

Contextual Notes

There are limitations regarding the assumptions about matrix properties, such as diagonalizability, and the effectiveness of the proposed methods may depend on specific cases or conditions.

sylar
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Today in the lecture we have learned the Putzer Algorithm, which enables us to calculate the higher powers(A^n, n>1 an integer) of a square matrix A. The other well-known methods use the Cayley-Hamilton Theorem, and the Jordan form of a matrix. I wonder whether there are other efficient ways to evaluate A^n.
 
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i don't know those algos but obviously you can diagonlize the matrix and exponentiate it like that
 
YOu can't always diagonalize a matrix. That was what he meant by "the Jordan form", for which diagonalizable matrices are a special case.
 
you can use QR decompisition to get a good approximation
 

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