📂Matrix Algebra

Pseudo Inverse Matrix

Overview

The Pseudoinverse Matrix generalizes the concept of an inverse matrix to matrices $A \in \mathbb{R}^{m \times n}$ that are not square matrices because their rows and columns are not of the same size. It refers to a matrix that acts as an ’effective’ inverse matrix for them. If a matrix transformation $T_{A} : \mathcal{N} (A) \to \mathcal{C} (A)$ satisfies
$$T_{A} \mathbf{x} = A \mathbf{x}$$ for all $\mathbf{x} \in \mathcal{N} (A)^{\perp}$, then $T_{A}$ becomes bijective. This can be seen as narrowing down the codomain of $T_{A}$ to forcefully make it surjective, and an inverse transformation $T_{A}^{-1} : \mathcal{C} (A) \to \mathcal{N} (A)$ exists through which the pseudoinverse can be defined.

Definition¹

Given a matrix $A \in \mathbb{R}^{m \times n}$ and matrix transformation $T_{A} : \mathcal{N} (A) \to \mathcal{C} (A)$, for every vector $\mathbf{y} = \mathbf{y}_{1} + \mathbf{y}_{2} \in \mathbb{R}^{m}$ if the matrix $A^{\dagger}$ satisfies the following, $A^{\dagger}$ is called the Moore-Penrose Pseudoinverse of $A$. $$A^{\dagger} \mathbf{y} = T_{A}^{-1} \mathbf{y}_{1}$$

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• Where $\mathbf{y}_{1} \in \mathcal{C}(A)$ and $\mathbf{y}_{2} \in \mathcal{C}(A)^{\perp}$.
• $\dagger$ is represented by a symbol that looks like a dagger pointed downwards, and indeed, it is read as Dagger. In mathematical physics, it might also represent a conjugate transpose matrix, but the context should help prevent confusion.

Theorem

The pseudoinverse of a matrix $A \in \mathbb{R}^{m \times n}$ is calculated as follows. $$\begin{align*} A^{\dagger} =& \lim_{\delta \to 0} \left( A^{T} A + \delta^{2} I \right)^{-1} A^{T} \\ =& \lim_{\delta \to 0} A^{T} \left( A^{T} A + \delta^{2} I \right)^{-1} \end{align*}$$