7.2 Self-adjoint transformations and the spectral decomposition theorem

Besides orthogonal transformations, there is another important class: transformations whose adjoints are themselves. Bear with me a bit, and we’ll see an example soon.

Definition 27. (Self-adjoint transformations)

Let f : ℝⁿ →ℝⁿ be a linear transformation. f is self-adjoint if f^∗ = f, that is,

⟨f(x), y⟩ = ⟨x, f(y)⟩

(7.3)

holds for all x,y ∈ℝⁿ.

As always, we are going to translate this into the language of matrices. If A is the matrix of f in the standard orthonormal basis, we know that A^T is the matrix of the adjoint. For self-adjoint transformations, it implies that A^T = A. Matrices such as these are called symmetric, and they have a lot of pleasant properties.

For us, the most important one is that symmetric matrices can be diagonalized! (That is,they can be transformed into a diagonal matrix with a check...