Can the system response function be calculated?

[Adel Makram]

Suppose we represent the input information as a (nx1) column vector, the output information as another (nx1) column vector and the system response function as a (nxn) matrix. My question, is it possible to calculate the values of the cells of the matrix knowing the input and the output?

For example, if the known values of the input vector are multiplied by the first row of the matrix, we will get the first value of the output vector which is already known. To solve for n-values of the first row of the response function matrix, we need to repeat this process n-times using n-different values of inputs and outputs. Will this be possible?

 

[FactChecker]

Yes. You would have to make sure that your n input vectors were linearly independent. The simple set of inputs where only one input index is nonzero at a time would give you an easy solution.

 

[Adel Makram]

Yes. You would have to make sure that your n input vectors were linearly independent. The simple set of inputs where only one input index is nonzero at a time would give you an easy solution.

In my mind, the linear dependency depends on the values of the matrix because the inputs can be chosen arbitrarily. But the values of matrix cells are themselves unknown, so how to make sure that the matrix coefficients can maintain linear independent sets of equations before being calculated?

 

[FactChecker]

Column m of matrix A is determined by Y = AX, where the column vector X is all zeros except the m'th element. a_i.m = y_i/x_m

 

[Adel Makram]

Column m of matrix A is determined by Y = AX, where the column vector X is all zeros except the m'th element. a_i.m = y_i/x_m

This is a good and simple method to calculate all a_i.m. But still it can not grantee that rows and columns of A are not linearly dependent.
Suppose for simplicity, that A is (2x2) matrix. For A to be diagonalizable, the following condition must be satisfied; a₂₁/a₁₁ ≠a₂₂/a₁₂.
So no matter which way we use to calculate a_im, the linearly dependence of rows and columns of A depends on the values of its elements.

 

[WWGD]

<Snip>
So no matter which way we use to calculate a_im, the linearly dependence of rows and columns of A depends on the values of its elements.

Of course, this is tautological.

 

[FactChecker]

There is no reason that the math model of a physical system must have outputs that are linearly independent. The outputs can be a, b, and c=a+b. The model can still be valid.

 

[WWGD]

One can always try to do linear regression, and if it does not work, e.g., r^2 is small, look for other types of regression.

 

[Adel Makram]

One can always try to do linear regression, and if it does not work, e.g., r^2 is small, look for other types of regression.

Do you mean multivariate linear regression like Y=XB, with Y is a random vector, B is a regressor vector and X is a matrix? Can you explain more please?

 

[WWGD]

Yes, sorry for the delay, multilinear regression in the sense you described, if the hypothesis for a linear regression being a good model hold.

 

[Adel Makram]

Yes, sorry for the delay, multilinear regression in the sense you described, if the hypothesis for a linear regression being a good model hold.

But in linear regression, we seek to calculate the regressors β₀ and β₁ by using different x_ij as representing χ matrix of independent variables. In my example, I am doing the opposite by seeking calculation of the system response function represented by matrix, χ in analogue with linear regression model.

 

[WWGD]

A, yes, sorry, let me rethink. I was thinking of least squares in a more general (maybe different) sense. Let me rethink.