Diagonal matrices and their properties

• When all the non-diagonal entries are zero.
• Represented as $\text{diag}(d_{11},...,d_{nn})$ since the only entries that matter are the ones on the diagonal.

Properties

1. Sum: since the non-diagonal entries are zero, we are basically doing $d_{nn}+c_{nn}$.
2. Product: looks very nice. Computing the mat-vec product, then this makes sense.
3. Power: raise each entry to $k$. Extends the product property from above.
   1. If $D$ is invertible, this property works even for negative powers.
   2. For $D$ to be invertible, all the diagonal entries must be non-zero (why?).
      1. For inverse, the determinant must not be zero cuz $A^{-1}=\frac{\text{adj }A}{\det A}$
      2. Det $\ne$ 0; for that the diagonal entries must not be zero.

Why Care?

• Simple computation.
• Useful to understand matrix properties. All the stuff happens across the main axis.