**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**.

**Solution**

Let **A** = **B** + i**C**, 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 z_{1} = y_{1} - iy_{n+1}, z_{2} = y_{2} - iy_{n+2} etc. Then the real part of **zA** is **x** as required.

© John Scholes

jscholes@kalva.demon.co.uk

7 Jan 2001