Cory Patrick

2022-06-25

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 $A{\overrightarrow{x}}_{i}\cdot {\overrightarrow{h}}_{j}$ for given ${\overrightarrow{x}}_{i}$ and ${\overrightarrow{h}}_{j}$

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 $A{\overrightarrow{x}}_{i}\cdot {\overrightarrow{h}}_{j}$ for given ${\overrightarrow{x}}_{i}$ and ${\overrightarrow{h}}_{j}$

Reagan Madden

Beginner2022-06-26Added 15 answers

Using unit vectors for both parts of the measurement you have ${e}_{i}^{T}A{e}_{j}={A}_{ij}$ so if “one measurement” corresponds to one unique pair $({\overrightarrow{x}}_{i},{\overrightarrow{h}}_{j})$ then each measurement of this form will yield one entry of the matrix. For a generic $m\times n$ matrix you'd therefore need $m\cdot n$ measurements. Knowing the trace adds one linear constraint, so you can infer one value from all the others, i.e. $m\cdot n-1$ measurements. I can not imagine how a different choice of measurement vectors could do better than this.

On the other hand, where I wrote $({\overrightarrow{x}}_{i},{\overrightarrow{h}}_{j})$ you had $({\overrightarrow{x}}_{i},{\overrightarrow{h}}_{j})$. Were you thinking of combining a single $x$ vector with multiple $h$ vectors, or vice versa? If so, you will need $m$ vectors ${\overrightarrow{h}}_{j}$ and $n$ vectors ${\overrightarrow{x}}_{i}$, minus one combination due to the trace. I'll leave it to you to decide what you count as “one measurement” in this case.

On the other hand, where I wrote $({\overrightarrow{x}}_{i},{\overrightarrow{h}}_{j})$ you had $({\overrightarrow{x}}_{i},{\overrightarrow{h}}_{j})$. Were you thinking of combining a single $x$ vector with multiple $h$ vectors, or vice versa? If so, you will need $m$ vectors ${\overrightarrow{h}}_{j}$ and $n$ vectors ${\overrightarrow{x}}_{i}$, minus one combination due to the trace. I'll leave it to you to decide what you count as “one measurement” in this case.

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