Does this norm inequality hold for projections onto the range of a sum of matrices? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:15:01Z http://mathoverflow.net/feeds/question/63724 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63724/does-this-norm-inequality-hold-for-projections-onto-the-range-of-a-sum-of-matrice Does this norm inequality hold for projections onto the range of a sum of matrices? Alex Gittens 2011-05-02T18:59:37Z 2011-05-02T20:47:09Z <p>Although it's simply stated, this is neither a homework problem or trivial (I think, but I'd be happy to be proven wrong :) ).</p> <p>Let $A,B$ be matrices and $x$ be a vector. Is it true that $$\|P_{A+B} x\| \geq \|P_A x\| - \|P_B x\|,$$ where $P_A$ is the projection onto the range space of $A$? (or is it true if you square the norms?)</p> <p>I'm having difficulty even figuring out how to attack this: every attempt I've made falters on the facts that the range space of $A + B$ is not simply related to those of $A$ and $B$ and that the projection is nonlinear. Random instances haven't yet provided counterexamples to the inequality.</p> http://mathoverflow.net/questions/63724/does-this-norm-inequality-hold-for-projections-onto-the-range-of-a-sum-of-matrice/63733#63733 Answer by Kate Juschenko for Does this norm inequality hold for projections onto the range of a sum of matrices? Kate Juschenko 2011-05-02T19:43:53Z 2011-05-02T20:47:09Z <p>by substituting $A$ with $A-B$ and $B$ by $-B$ your condition will be equivalent to the following: <code>$||P_{A+B}x||\leq ||P_Ax||+||P_Bx||.$</code> the last one is not true in general (for non-positive matrices), for example it does not hold for $A=((0,1),(0,0))$, $B=((1,1),(1,1))$ and $x=\frac{1}{\sqrt{2}}(1,-1)$.</p> http://mathoverflow.net/questions/63724/does-this-norm-inequality-hold-for-projections-onto-the-range-of-a-sum-of-matrice/63734#63734 Answer by Michael Renardy for Does this norm inequality hold for projections onto the range of a sum of matrices? Michael Renardy 2011-05-02T19:49:31Z 2011-05-02T19:49:31Z <p>Let $$A=\pmatrix{1&amp;0\cr 0&amp;0}, B=\pmatrix{0&amp;0\cr 1&amp;0}, x=(x_1,x_2).$$ Then $P_Ax=(x_1,0)$, $P_Bx=(0,x_2)$, $P_{A+B}x=((x_1+x_2)/2,(x_1+x_2)/2)$. Thus you are asking if $$|(x_1+x_2)/\sqrt{2}|\ge |x_1|-|x_2|.$$ Clearly, this is false in general. </p>