Case of equality in spectral norm matrix triangle inequality

Question

Let $(A,B)\in M_n(\mathbb{R})\times M_n(\mathbb{R})$ be two matrices. We denote by $\|\cdot\|_2$ the spectral norm. Without any additional assumptions on $A$ and $B$, can we characterize the case of equality in the spectral norm matrix triangle inequality, i.e., when \begin{align*} \|A+B\|_2=\|A\|_2+\|B\|_2. \end{align*} If $A=U\Sigma_AV^T$ and $B=U\Sigma_BV^T$ (singular value decompositions), the equality holds but is there a full characterization? Best wishes.

Answer 1

Equality holds if and only if $A$ and $B$ share a common right singular vector $v$ corresponding to the largest singular value and $Av=\sigma_1(A)u,\,Bv=\sigma_1(B)u$ for the same left singular vector $u$.

To prove this, suppose equality holds. If $\|A\|_2=\|B\|_2=0$, there is nothing to prove. So, suppose also that $A,B\ne0$. Let $v$ be a right singular vector of $A+B$ corresponding to the largest singular value. Then $$ \begin{align} \|(A+B)v\|_2 &\le\|Av\|_2+\|Bv\|_2\tag{1}\\ &\le\max_{\|x\|_2=1}\|Ax\|_2+\max_{\|y\|_2=1}\|Ay\|_2\tag{2}\\ &=\|A\|_2+\|B\|_2\\ &=\|A+B\|_2\\ &=\|(A+B)v\|_2. \end{align} $$ Therefore ties occur in both $(1)$ and $(2)$. The tie that occurs in $(2)$ means that $v$ is also a unit right singular vector of both $A$ and $B$ corresponding to their largest singular values. Therefore $Av=\sigma_1(A)u$ and $Bv=\sigma_1(B)u'$ for some unit left singular vectors $u$ and $u'$ of $A$ and $B$ respectively. However, the tie in the triangle inequality $(1)$ implies that $Av$ and $Bv$ are real positive multiples of each other. Therefore $u=u'$ and we are done.

Using the characterisation $\|A\|_2=\max_{\|x\|_2=\|y\|_2=1}x^TAy$, the necessary and sufficient condition above can be rephrased as $$ \|A\|_2=u^TAv\quad\text{and}\quad\|B\|_2=u^TBv $$ for some unit vectors $u$ and $v$. This is also equivalent to the condition that there exist two orthogonal matrices $U$ and $V$ such that $$ U^TAV=\pmatrix{\sigma_1(A)\\ &A'} \quad\text{and}\quad U^TBV=\pmatrix{\sigma_1(B)\\ &B'} $$ for some matrices $A'$ and $B'$.