I know that the spectral theorem tells us that in the case of a real inner product space, an operator is self adjoint if and only if there is an orthonormal basis with only eigenvectors of that operator and that in the case of a complex inner product space, an operator is normal if and only if there is an orthonormal basis with only eigenvectors.
However, I am unable to see what is so important about this. I know that this means I can diagonalize the matrix but I don't know what is so good about diagonalizing a matrix. Also, why do we have to have an orthonormal basis? Why can it not be a standard basis?
Thank you!
Best Answer
Write down a square matrix, $A$. Now, raise it to the power 100. Not so easy, is it? Well, it is if the matrix is diagonal. It's also easy if the matrix is diagonalizable; if $P^{-1}AP=D$ is diagonal, then $A^{100}=PD^{100}P^{-1}$. So, computing high powers of matrices is made easy by diagonalization.
And why would you want to compute high powers of a matrix? Well, many things are modelled by discrete linear dynamical systems, which is a fancy way of saying you have a sequence of vectors $v_0,v_1,v_2,\dots$ where you get each vector (after the first) by multiplying the previous vector by $A$. But then $v_k=A^kv_0$, and voila! there's your high power of a matrix.