5.10 Eigenvectors and eigenvalues

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

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, 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 zeros. Some of the d_j might be zero.

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_j are 0.
• If D is invertible,

• If {b₁, b₂, …...