----- > [!proposition] Proposition. ([[sufficiency conditions of invertibility of gram matrix sum]]) > 1. $A'A + B'B$ is [[inverse matrix|invertible]] if $B$ has [[linearly independent]] columns. > 2. $A'A + B'B$ is [[inverse matrix|invertible]] if $\ker A \cap \ker B = \{ 0 \}$. > > [!proof]- Proof. ([[sufficiency conditions of invertibility of gram matrix sum]]) > 1. By [[columns linearly independent iff gram matrix is invertible]], $B'B$ is [[inverse matrix|invertible]] and thus has trivial [[kernel]]; since [[Gram matrices are positive semidefinite]] this means all its [[eigenvalue]]s are positive. Thus $B'B$ is [[positive definite matrix|positive definite]]. Now, by [[sum of positive definite and positive semidefinite matrix is positive definite]] we see $A'A + B'B$ is [[positive definite matrix|positive definite]]; in particular, this means its $0$-eigenspace has dimension $0$. It is therefore invertible, because it has a trivial [[kernel]]. > 2. We will show the contrapositive. Suppose $A'A + B'B$ is not [[inverse matrix|invertible]]. Then by [[sum of positive definite and positive semidefinite matrix is positive definite]] and the fact that [[positive definite matrix|positive definite]] [[matrix|matrices]] are [[inverse matrix|invertible]], it must be true that $A'A+B'B$ is PSD but not PD. Hence there exists nonzero $x$ such that $x'(A'A + B'B)x=x'A'Ax+x'B'Bx=0$. But both $A'A$ and $B'B$ are [[positive semidefinite matrix|PSD]], so this can only if and only if $x'A'Ax=0=x'B'Bx$, which in turn happens if and only if $\langle Ax,Ax \rangle=0=\langle Bx,Bx \rangle$ where $\langle \cdot , \cdot\rangle$ denotes the [[Euclidean inner product]]. The only [[vector]] [[orthogonal]] to itself is the zero vector, so $Ax=Bx=0$ and we have a nontrivial $x \in \ker A \cap \ker B$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```