A is a complex 2n x n matrix such that if z is a real 1 x 2n row vector then z A ≠ 0 unless z = 0. Prove that given any real 2n x 1 column vector x we can always find a complex n x 1 column vector z such that the real part of A z = x.
Let A = B + iC, where B and C are real matrices. Let D be the 2n x 2n real matrix (B C). Then since there is no non-zero real row vector z such that zA = 0, it follows that there is no non-zero real row vector z such that zD =0. In other words D is invertible. So given any real 2n x 1 column vector x we can find a real 2n x 1 column vector y such that Dy = x. Now define a complex n x 1 column vector z by z1 = y1 - iyn+1, z2 = y2 - iyn+2 etc. Then the real part of zA is x as required.
48th Putnam 1987
© John Scholes
7 Jan 2001