Matrix Exponentiation with Diagonalization

Raise the matrix $A = \begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix}$ to the 100th power using diagonalization.

Raising a matrix to a large power using diagonalization is a technique that leverages the properties of eigenvalues and eigenvectors. This approach is particularly efficient compared to standard matrix multiplication, which becomes computationally expensive for large powers. The key here is to decompose the matrix into its diagonal form, which simplifies the exponentiation process considerably.

Diagonalization is applicable to square matrices that can be expressed as a product of their eigenvectors and a diagonal matrix of their eigenvalues. Diagonal matrices are simple to exponentiate because raising a diagonal matrix to a power is achieved by raising each of its diagonal elements to that power individually. In essence, the problem transforms into calculating powers of scalars rather than matrices. This process significantly reduces the computational complexity and highlights the elegance of linear algebra techniques in solving complex problems efficiently.