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

$$\rank(AB) \leq \rank(A), \quad \rank (AB) \leq \rank(B)$$