The discussion focuses on finding formulas for the entries of the matrix M raised to the power of n, where M is defined as M = [8, -1; 2, 11]. The solution involves diagonalization, where D is the transformation matrix that diagonalizes M, leading to the relationship Mn = D-1 ᴏMn D. The diagonal entries of the diagonalized matrix ᴏM represent the eigenvalues of M, while the columns of D correspond to the eigenvectors of M.
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Also, The diagonal entries in [itex]\tilde{M}[/itex] are the eigenvalues of the matrix M and the columns of D are the corresponding eigenvectors of M.clamtrox said:Let's suppose D is the transform that diagonalizes M, and denote [itex]D M D^{-1} = \tilde{M}[/itex]. Then
[tex]M^n = M M M ... M = D^{-1} D M D^{-1} D M D^{-1} D M ... M D^{-1} D = D^{-1} \tilde{M}^n D.[/tex]