I have a matrix A with null trace.What is the minimum number of linear measurements that I have to perform in order to determine A?By a linear measurement I mean that I know the quantities A x → i ⋅ h → j for given x → i and h → j

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I have a matrix   A with null trace.What is the minimum number of linear measurements that I have to perform in order to determine   A?By a linear measurement I mean that I know the quantities   A                    x        →              i    ⋅                    h        →              j   for given                     x        →              i   and                     h        →              j

Reagan Madden · Accepted Answer

Using unit vectors for both parts of the measurement you have       e    i    T    A      e    j    =      A          i      j       so if “one measurement” corresponds to one unique pair   (                    x        →              i    ,                    h        →              j    ) then each measurement of this form will yield one entry of the matrix. For a generic   m  ×  n matrix you&#039;d therefore need   m  ⋅  n measurements. Knowing the trace adds one linear constraint, so you can infer one value from all the others, i.e.   m  ⋅  n  −  1 measurements. I can not imagine how a different choice of measurement vectors could do better than this.On the other hand, where I wrote   (                    x        →              i    ,                    h        →              j    ) you had   (                    x        →              i    ,                    h        →              j    ). Were you thinking of combining a single   x vector with multiple   h vectors, or vice versa? If so, you will need   m vectors                     h        →              j   and   n vectors                     x        →              i  , minus one combination due to the trace. I&#039;ll leave it to you to decide what you count as “one measurement” in this case.

I have a matrix A with null trace. What is the minimum number of linear measurements that I hav

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