Can the system response function be calculated?

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
Gold Member
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.

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
Gold Member
Column m of matrix A is determined by Y = AX, where the column vector X is all zeros except the m'th element. ai.m = yi/xm

Column m of matrix A is determined by Y = AX, where the column vector X is all zeros except the m'th element. ai.m = yi/xm
This is a good and simple method to calculate all ai.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 aim, the linearly dependence of rows and columns of A depends on the values of its elements.

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

Of course, this is tautological.

FactChecker
Gold Member
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
Gold Member
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.

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