--- > [!proposition]+ Proposition. ([[additivity condition of the nuclear norm]]) > Let $A,B$ have the same size. If $AB'=0$ and $A'B=0$ then $\|A+B\|_{*}=\|A\|_{*}+\|B\|_{*},$ > where $\|\cdot\|_{*}$ denotes the [[nuclear norm]]. ^proposition > [!proof]+ Proof. ([[additivity condition of the nuclear norm]]) > ~ > > Write [[compact svd]]s $\begin{align} A = & U_{r} \Sigma_{r} V_{r}' \ \text{ where } r=\text{rank}A \\ B = & X_{s} \Omega_{s}Y_{s}', \ \text{where } s= \text{rank} B. \end{align}$ > Observe that $AB'=0$ implies $U_{r}\Sigma_{r}V_{r}' Y_{s}\Omega_{s}X_{s}=0$. In particular, since all matrices in the product are [[inverse matrix|invertible]] we can left and right multiply both sides by relevant inverses to see $V_{r}'Y_{s}=0$. > > $\begin{align} ((A'A+B'B)^{1/2})^{2}= & A'A+B'B \\ = & A'A + B'B + 0 \\ = & A'A + B'B + \overbrace{V_{r}'Y_{s}}^{=0}\\ = & A'A + B'B + \overbrace{V_{r}\Sigma_{r}^{}V_{r}'Y_{s}\Omega_{s}^{}Y_{s}' }^{=0} \\ = & A'A + B'B + {\sqrt{ (V_{r}\Sigma_{r}^{}V_{r}')^{2}}^{}\sqrt{ (Y_{s}\Omega_{s}^{}Y_{s}')^{2}}^{} }\\ = & A'A + B'B + \sqrt{ V_{r} \Sigma_{r}^{2} V_{r}' } \sqrt{ Y_{s} \Omega_{s}^{2} Y_{s}'} \\ = & A'A + B'B + (A'A)^{1/2} (B'B)^{1/2} + \overbrace{(B'B)^{1/2} (A'A)^{1/2} }^{=0 \text{ (same reasoning as left term)}} \\ = & ((A'A)^{1/2} + (B'B) ^{1/2})^{2}. \end{align}$ > > $A'A$ and $B'B$ are are [[positive semidefinite matrix|PSD]] ([[Gram matrices are positive semidefinite]]) and $A'A+B'B$ is a [[sum of positive semidefinite matrices is positive semidefinite|sum of two positive definite matrices]] and thus [[positive semidefinite matrix|PSD]]. [[Every positive semidefinite operator has exactly one positive square root]], so we conclude that $(A'A+B'B)^{1/2}=(A'A)^{1/2}+(B'B)^{1/2},$ > from which it follow that $\begin{align} \|A+B\|_{*} = & \text{Tr } ((A+B)'(A+B))^{1/2} \\ = & \text{Tr }(A'A+B'A +B'B+A'B)^{1/2} \\ = & \text{Tr }(A'A+B'B)^{1/2} \\ = & \text{Tr } ( \ (A'A)^{1/2} + (B'B)^{1/2} \ ) \\ = & \text{Tr } (A'A)^{1/2} + \text{Tr }(B'B)^{1/2} \ \\ = & \|A\|_{*} + \|B\|_{*}. \end{align}$ ^proof --- #### 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 > ```