5.9 Eigenvectors and eigenvalues

Let’s pause to review some of the wonderful features of diagonal matrices. Recall that a diagonal matrix has 0s everywhere except maybe on the main diagonal. A simple example for R³ is

Its effect on the standard basis vectors e₁, e₂, and e₃ is to stretch by a factor of 3 along the first, leave the second alone, and reflect across the xy-plane and then stretch by a factor of 2 along the third. A general diagonal matrix looks like

Of course, we might be dealing with a small matrix and not have quite so many 0s. For a diagonal matrix D as above,

• det(D) = d₁ d₂ ··· d_n.
• tr (D) = d₁ + d₂ + ··· + d_n
• D^T = D.
• D is invertible if and only if none of the d_i are 0.
• If D is invertible,
  
• If b₁, ..., b_n is the basis we are using, then D b₁ = d₁ b₁, D b₂ = d₂ b₂, ..., D b_n = d_n b_n.

Focusing on this last...