Representation of the diagonal: Difference between revisions
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For all <math>C > 0</math> there exists a diagonal matrix with trace at least <math>C</math> that can be expressed as <math>\sum_i \lambda_i P_i \otimes Q_i</math>, where <math>\sum_i | \lambda_i | | For all <math>C > 0</math> there exists a diagonal matrix with trace at least <math>C</math> that can be expressed as <math>\sum_i \lambda_i P_i \otimes Q_i</math>, where <math>\sum_i | \lambda_i | \leq 1</math> and each <math>P_i</math> and <math>Q_i</math> is the characteristic function of a HAP. | ||
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== Proof of implication == | == Proof of implication == | ||
Suppose <math>D</math> is a diagonal matrix with entries <math>b_j</math> on the diagonal, with <math>\sum_j b_j</math> unbounded, and <math>D = \sum_i \lambda_i P_i \otimes Q_i</math> where <math>\sum_i | \lambda_i | | Suppose <math>D</math> is a diagonal matrix with entries <math>b_j</math> on the diagonal, with <math>\sum_j b_j</math> unbounded, and <math>D = \sum_i \lambda_i P_i \otimes Q_i</math> where <math>\sum_i | \lambda_i | \leq 1</math> and the <math>P_i</math> and <math>Q_i</math> are HAPs. Suppose <math>x</math> is a <math>\pm 1</math> sequence with finite discrepancy <math>C</math>. Then we can write <math>| \langle x, Dx \rangle |</math> in two ways. On the one hand, <math>| \langle x, Dx \rangle | = | \sum_j b_j x_j^2 | = | \sum_j b_j |</math>, which is unbounded; on the other hand, <math>| \langle x, Dx \rangle | = | \langle x, \sum_i \lambda_i (P_i \otimes Q_i) x \rangle | = | \sum_i \lambda_i \langle x, P_i \rangle \langle x, Q_i \rangle | \leq \sum_i | \lambda_i | | \langle x, P_i \rangle | | \langle x, Q_i \rangle | \leq C^2</math>, a contradiction. | ||
== Possible proof strategies == | == Possible proof strategies == | ||
Revision as of 20:32, 30 April 2010
The following conjecture, if true, would imply the Erdos discrepancy conjecture.
For all [math]\displaystyle{ C \gt 0 }[/math] there exists a diagonal matrix with trace at least [math]\displaystyle{ C }[/math] that can be expressed as [math]\displaystyle{ \sum_i \lambda_i P_i \otimes Q_i }[/math], where [math]\displaystyle{ \sum_i | \lambda_i | \leq 1 }[/math] and each [math]\displaystyle{ P_i }[/math] and [math]\displaystyle{ Q_i }[/math] is the characteristic function of a HAP.
Proof of implication
Suppose [math]\displaystyle{ D }[/math] is a diagonal matrix with entries [math]\displaystyle{ b_j }[/math] on the diagonal, with [math]\displaystyle{ \sum_j b_j }[/math] unbounded, and [math]\displaystyle{ D = \sum_i \lambda_i P_i \otimes Q_i }[/math] where [math]\displaystyle{ \sum_i | \lambda_i | \leq 1 }[/math] and the [math]\displaystyle{ P_i }[/math] and [math]\displaystyle{ Q_i }[/math] are HAPs. Suppose [math]\displaystyle{ x }[/math] is a [math]\displaystyle{ \pm 1 }[/math] sequence with finite discrepancy [math]\displaystyle{ C }[/math]. Then we can write [math]\displaystyle{ | \langle x, Dx \rangle | }[/math] in two ways. On the one hand, [math]\displaystyle{ | \langle x, Dx \rangle | = | \sum_j b_j x_j^2 | = | \sum_j b_j | }[/math], which is unbounded; on the other hand, [math]\displaystyle{ | \langle x, Dx \rangle | = | \langle x, \sum_i \lambda_i (P_i \otimes Q_i) x \rangle | = | \sum_i \lambda_i \langle x, P_i \rangle \langle x, Q_i \rangle | \leq \sum_i | \lambda_i | | \langle x, P_i \rangle | | \langle x, Q_i \rangle | \leq C^2 }[/math], a contradiction.
