- #1

synMehdi

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- TL;DR Summary
- I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space.

**Summary:**I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space.

I need to do a linear model identification using least squared method.

My model to identify is a matrix ##[A]##. My linear system in the complex space is:

$$[A]_{_{n \times m}} \cdot [x]_{_{m \times 1}} = [y]_{_{n \times 1}}$$

where ##n## and ##m## define the matrix sizes. in my notation I define my known arrays ##x## and ##y## as vectors.

To identify ##[A]## I have a set of ##p## equations:

$$[A] \cdot \vec{x}_1 = \vec{y}_1$$

$$[A] \cdot \vec{x}_2 = \vec{y}_2$$

$$...$$

$$[A] \cdot \vec{x}_p = \vec{y}_p$$

knowing that my system is overdetermined (##p>n,m##) and that each pair of ##\vec{x}## and ##\vec{y}##, is known, I want to identify my linear model matrix ##[A]## with least squares.

**My approach:**

I have aranged my known equations like above:

$$[A] \cdot [\vec{x}_1\>\vec{x}_2\>...\>\vec{x}_p]=[\vec{y}_1\>\vec{y}_2\>...\>\vec{y}_p]$$

My initial linear system becomes a matrix equation:

$$[A]_{_{n \times m}} \cdot [X]_{_{m \times p}} = [Y]_{_{n \times p}}$$

Is this the right thing to do to solve ##[A]## with the Moore-Penrose inverse of ##[X]##? What is the correct way to do this?