Section 7.5 Theory
Here we record properties of these special forms of matrices relative to the various matrix operations.
Subsection 7.5.1 Algebra of special forms
First, we summarize some of the algebra of working with these forms. We have already explored proving parts of the proposition below in Discovery 7.5 and Discovery 7.6, so below we provide similar proofs for a couple more parts.
Proposition 7.5.1.
- The result of adding two diagonal matrices, scalar multiplying a diagonal matrix, multiplying two diagonal matrices, taking an inverse of a diagonal matrix, or taking a power (positive or negative) of a diagonal matrix is always a diagonal matrix.
- Statement 1 remains true
- when all occurrences of the word “diagonal” are replaced by “scalar,” or
- when all occurrences of the word “diagonal” are replaced by “upper triangular,” or
- when all occurrences of “diagonal” are replaced by “lower triangular.”
- Statement 1 remains true when all occurrences of the word “diagonal” are replaced by “symmetric,” except that the product of two symmetric matrices may not be symmetric.
Partial proof of Statement 2.
We will prove that the result of scalar multiplying an upper triangular matrix is again upper triangular. As we discovered in Discovery 7.1.g, an upper triangular matrix \(U\) is characterized by having all entries \(u_{ij}\) equal to \(0\) for \(i>j\) (i.e. entries below the main diagonal). The scalar multiple \(kU\) has entries \([kU]_{ij} = ku_{ij}\text{,}\) so if \(u_{ij} = 0\) for \(i>j\text{,}\) then also \(k u_{ij} = 0\) for \(i>j\text{,}\) and the matrix \(kU\) is also upper triangular.
Partial proof of Statement 3.
We will prove that the inverse of an invertible, symmetric matrix is again symmetric. So suppose that \(A\) is both invertible and symmetric. By definition of symmetry, this means that \(A\) is equal to its own transpose. We would like to verify that \(\inv{A}\) is also symmetric; that is, that \(\inv{A}\) is equal to its own transpose. Let’s do that, using proper LHS vs RHS procedure for the proposed equality \(\utrans{(\inv{A})} = \inv{A}\text{:}\)
\begin{align*}
\text{LHS} \amp = \utrans{(\inv{A})}\\
\amp = \inv{(\utrans{A})} \amp \amp\text{(i)}\\
\amp= \inv{(A)} \amp \amp\text{(ii)}\\
\amp= \text{RHS},
\end{align*}
with justifications
- Proposition 5.5.8; and
- \(\utrans{A}=A\) by symmetric assumption.
Subsection 7.5.2 Invertibility of special forms
Finally, we record our observations about the invertibility of some of these special forms. The following fact was already discussed in Subsection 7.3.2, so we will not formally prove it.
Proposition 7.5.2.
An upper or lower triangular matrix is invertible precisely when the entries on its main diagonal are all nonzero.
