# Consider the complex-valued matrix

HW Problem 12
Consider the complex-valued matrix
2 3+ jc —8 a+ jb
V = [ v(1) v(2) v(3) v(4) ] = a + jb 2 3 + jc —8
—8 a + jb 2 3 + jc
3+ jc —8 a+ jb 2 (i) (6 pts.) Show that there exists only one choice of constants a a R, b a R and c > 0 such that
the columns of V are pairwise orthogonal. For that choice of a, b and c, what are the resulting
column norms? ( You will need to set two column inner products equal to zero. Check your answers
in MATLAB using V’ *V before proceeding further.)
From now on, assume that a, b, c and d are as found in part (i) above.
(ii) (6 pts.) Determine d such that the real-valued vector
s = [ 27 45 41 23 1T
equals Vd. (Gaussian elimination is not needed here. Again, verify your answers in MATLAB.)
(iii) (5 pts.) Determine the projection b of s onto the subspace generated by the vectors v(2) and
v(4). What is the value of Ms — 42?
(iv) (3 pts.) If

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