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Putnam 1987 B5

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August 5, 2019
Putnamlinear algebramatrix

Problem Statement

Let OnO_n be the nn-dimensional vector (0,0,⋯ ,0)(0,0,\cdots, 0). Let MM be a 2n×n2n \times n matrix of complex numbers such that whenever (z1,z2,…,z2n)M=On(z_1, z_2, \dots, z_{2n})M = O_n, with complex ziz_i, not all zero, then at least one of the ziz_i is not real. Prove that for arbitrary real numbers r1,r2,…,r2nr_1, r_2, \dots, r_{2n}, there are complex numbers w1,w2,…,wnw_1, w_2, \dots, w_n such that re[M(w1⋮wn)]=(r1⋮rn). \mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array} \right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n \end{array} \right). (Note: if CC is a matrix of complex numbers, re(C)\mathrm{re}(C) is the matrix whose entries are the real parts of the entries of CC.)
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