Find (in case there is any) which complex vector n of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix. n = [ a , b ] T = [ a 1 + j a 2 , b 1 + j b 2 ] T n n † = IObtain the following set of 4 equations: a 1 2 + a 2 2 = 1 b 1 2 + b 2 2 = 1 a 1 b 2 + a 2 b 1 = 0

Question

Find (in case there is any) which complex vector   n of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix.  n  =  [  a  ,  b      ]    T    =  [      a    1    +  j      a    2    ,      b    1    +  j      b    2        ]    T    n      n    †    =  IObtain the following set of 4 equations:      a    1    2    +      a    2    2    =  1      b    1    2    +      b    2    2    =  1      a    1        b    2    +      a    2        b    1    =  0

Blaine Day · Accepted Answer

There is no solution. Given a vector   n  ∈            C        2  ,   n      n          †       has rank (at most)   1, whereas the   2  ×  2 identity matrix has rank   2. One can produce diagonal matrices this way, but by this consideration, these will always have at most   1 nonzero entry.

Find (in case there is any) which complex vector n of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix. n=[a,b]^T=[a_1+ja_2,b_1+jb_2]^T nn^†=I Obtain the following set of 4 equations: a_2^1+a_2^2=1, b_2^1+b_2^2=1, a_1 b_2+a_2 b_1=0

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