General truth about elements of a matrix to the n-th power?

[nonequilibrium]

Say we have a matrix P with eigenvalues $\lambda_1, \cdots, \lambda_n$ (possibly some are the same) and P can be diagonalized, then we can always say that the element on the a'th row and b'th column of P^n is equal to $P^n(a,b) = \sum_{i = 1}^n \alpha_i \lambda_i^n$ with $\alpha_i$ independent of n (but dependent on a and b).

Correct? And I don't think the conditions can be weakened?

 

[micromass]

Seems indeed correct.

 

[nonequilibrium]

I find it to be rather pretty :)

 

[micromass]

Indeed, it IS pretty!

 

[CellCoree]

I can confirm that the statement is correct. This result is known as the diagonalization theorem and it states that any square matrix can be diagonalized if it has n distinct eigenvalues. This means that the matrix can be expressed as a product of a diagonal matrix and a matrix of eigenvectors.

In terms of the general truth about elements of a matrix to the n-th power, we can say that the element at position (a,b) of the matrix P^n can be calculated by taking the sum of the products of the eigenvalues raised to the n-th power and the corresponding elements of the eigenvector matrix. This result is independent of n, meaning that it holds true for any power of the matrix.

Furthermore, the conditions of the diagonalization theorem cannot be weakened. If a matrix does not have n distinct eigenvalues, it cannot be diagonalized and therefore the above result would not hold. This theorem is an important tool in understanding and analyzing matrices, and it has many applications in various fields of science and mathematics.