Why is the 2-norm of A^k equal to 2-norm of T^k?

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The equality of the 2-norms, ||A^k|| = ||T^k||, for k ≥ 0 is established through the Schur decomposition of matrix A, represented as Q^H * A * Q = T, where T consists of a diagonal matrix D and a strictly upper triangular matrix N. The diagonal matrix D contains the eigenvalues of A, which directly influences the 2-norm calculation. This relationship is crucial in understanding the behavior of matrix powers in linear algebra, particularly in spectral theory.

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Let Q^H * A * Q = T = D + N be Schur decomposition of A. D is a diagonal matrix with eigenvalues of A and N is strictly upper triangular matrix. This sentence is used in a proof which I am not going to state but at some point it states that ||A^k|| 2 norm = ||T^k|| 2 norm for k>=0. I don't understand
where this equality comes from. I tried to work this equality out with an example of taking
 
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The problem is simple: why is the 2-norm of A^k equal to 2-norm of T^k?
 

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