A problem concerning two symmetric matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:08:21Z http://mathoverflow.net/feeds/question/19822 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19822/a-problem-concerning-two-symmetric-matrices A problem concerning two symmetric matrices zhaoliang 2010-03-30T12:32:16Z 2010-04-19T18:47:33Z <p>Let A , B denote two symmetric matrices of the same order n. and Spec(A)=X , Spec(B)=Y.</p> <p>If Spec(A+B)=X <code>$\cup$</code> Y , proof thar AB=0.</p> <p>here Spec(A) means the set of the engevalues of A.</p> <p>This is a problem posed in a math forum bbs in China , I find it interesting, so I </p> <p>also put it here.</p> http://mathoverflow.net/questions/19822/a-problem-concerning-two-symmetric-matrices/19825#19825 Answer by Thomas Kragh for A problem concerning two symmetric matrices Thomas Kragh 2010-03-30T12:53:25Z 2010-03-30T12:53:25Z <p>Counter example:</p> <p>take the diagonal matrices:</p> <p>$A=D(2,1,1,0,0)$ and $B=D(0,0,1,2,0)$</p> <p>If you want the multiplicity to match (thinking of $X \cup Y$ is a "multiset" union) then it is easy to create an inductive argument proving the assertion.</p> http://mathoverflow.net/questions/19822/a-problem-concerning-two-symmetric-matrices/21872#21872 Answer by Portland for A problem concerning two symmetric matrices Portland 2010-04-19T18:47:33Z 2010-04-19T18:47:33Z <p>Zhaoliang, </p> <p>Maybe you wanted to ask this question:</p> <p>Let $A$ and $B$ be two $n \times n$ real symmetric matrices such that $$\det(I_n-xA)\det(I_n-yB) = \det(I_n - xA-yB)$$ holds for all real values of $x$ and $y$. Then $A B = 0$.</p> <p>There are many proof, my favorite is probably a short proof in the paper <em>On a matrix theorem of A. T. Craig and H. Hotelling</em> by Olga Taussky.</p> <p>You can also assume only that $\forall x\in \mathbb{R}, \det(I_n-xA)\det(I_n-xB) = \det(I_n - xA-xB)$, then you still have $AB=0$, but this is not in Taussky's article.</p> <p>For those of you interested, here is a variant: If $\mathcal{S}\subset \mathbb{R}$ such that $|\mathcal{S}|=n^2$, and $\forall x\in \mathcal{S}, \det(I_n-xA)\det(I_n-xB) = \det(I_n - xA-xB)$, do we necessarily have $AB=0$?</p>