Properties of Linear and Matrix Operators

Define the adjoint \(A^*\) of operator \(A\) such that
\[ \DeclareMathOperator{\rank}{rank} \langle y, Ax \rangle = \langle A^*y, x \rangle \]
We have the properties

• \(\mathcal{N}(A) = \mathcal{N}(A^*A)\) and \(\mathcal{R}(A^*) = \mathcal{R}(A^*A)\)
• \(\mathcal{N}(A^*) = \mathcal{N}(AA^*)\) and \(\mathcal{R}(A) = \mathcal{R}(AA^*)\)

And noting that \(\dim \mathcal{R}(A) = \dim \mathcal{R}(A^*)\), we have

• \(\rank(A^*A) = \rank ( AA^*) = \rank(A) = \rank(A^*) \)

For matrix operators, dimension of the column space is equal to the dimension of the row space

• column space: \(\dim (\mathcal{R}(A)) = r\)
• row space: \(\dim (\mathcal{R}(A^H)) = r\)
• Nullspace: \(\dim (\mathcal{N}(A)) = n -r\)
• Left nullspace: \(\dim (\mathcal{N}(A^H))= m-r\)

Characterization of matrix \(AB\)

For matrices A and B such that AB exists

1. \(\mathcal{N}(B) \subset \mathcal{N}(AB)\)
2. \(\mathcal{R}(AB) \subset \mathcal{R}(A)\)
3. \(\mathcal{N}(A^*) \subset \mathcal{N}((AB)^*)\)
4. \(\mathcal{R}((AB)^*) \subset \mathcal{R}(B^*)\)

From 2 and 4

\[ \rank(AB) \leq \rank(A), \quad \rank (AB) \leq \rank(B)  \]