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Can anyone help me to show if $Z_1$, $Z_2$, and $Z_3$ are projection matrices (i.e. idempotent and symmetric) and if $Z_1 + Z_2 + Z_3 = I_n$, then we can conclude:

$$\forall i \ne j \text{ ,} Z_i Z_j = 0$$

It seems to be an easy problem but I was not able to solve it.

Thanks,

Julien
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Sam
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  • $x(Z_1+Z_2+Z_3)=x$. So if the projection matrices were not orthogonal, you would be adding a component of $x$ to itself more than once and the expression wouldn't be true. The orthoganality of $Z_i$ ensures that each part of x is expressed uniquely. – Kieran Cooney Oct 28 '13 at 15:08
  • @KieranCooney Symmetry ensures that these are orthogonal projections – tfw cant into math Oct 28 '13 at 15:17
  • I proved a more general result at http://math.stackexchange.com/questions/507796/problem-with-sum-of-projections/525709#525709 – Ewan Delanoy Oct 28 '13 at 15:28
  • A slightly more general version was asked, and even answered on MO, so it is not so trivial, if you have never seen the trick (not sure it belongs in MO though). Note that you don't need orthogonality. Idempotency is enough. – Julien Oct 28 '13 at 15:34

2 Answers2

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Here is an argument that works for any number of projections. It uses only the algebraic properties of the projections (that they are selfadjoint and positive, and that they equal their own square), without using them as operators. Also, equality to the identity is not required, just that the sum is below the identity.

So, assume$$\sum_{k=1}^NZ_k\leq I.$$ Take indices $i$ and $j$ with $i\ne j$. Then $$ 0\leq Z_iZ_jZ_i\leq Z_i\left(\sum_{k\ne i} Z_k\right)Z_i\leq Z_i(I-Z_i)Z_i=0. $$ It follows that $Z_iZ_jZ_i=0$. But then $$ 0=Z_iZ_jZ_i=Z_iZ_j^2Z_i=(Z_jZ_i)^*Z_jZ_i, $$ and so $Z_jZ_i=0$.

Martin Argerami
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In fact if $P+Q$ is a projection, then $PQ = 0$, because $(P+Q)^2 = (P+Q)$, and so $$ PQ(x) + QP(x) = 0 \Rightarrow PQ = -QP $$ and hence for any $x \in ran(PQ)$, $$ PQx = x \Rightarrow Px = x $$ and similarly, $$ -QPx = x \Rightarrow Qx = x $$ and hence $x = -QPx = -Qx = -x$, and so $x = 0$, which means that $PQ = 0$

Now apply this to $P = Z_i, Q=Z_j$ for $i\neq j$

Prahlad Vaidyanathan
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