Definition:Pseudoinverse of Linear Transformation

Definition

Let $U, V$ be vector spaces.

Let $S: U \to V$ be a linear transformation.

Let $T: V \to U$ be a linear transformation.

$S$ and $T$ are pseudoinverse to each other if and only if:

$T \circ S - I_U$ is degenerate

and:

$S \circ T - I_V$ is degenerate

where:

$\circ$ denotes the composition

$I_U$ denotes the identity mapping of $U$

$I_V$ denotes the identity mapping of $V$

This article, or a section of it, needs explaining. In particular: $T \circ S - I_U$ and $S \circ T - I_V$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Also see

• Definition:Pseudoinverse of Bounded Linear Transformation
• Linear Transformation is Fredholm Operator iff Pseudoinverse exists

Sources

• 2002: Peter D. Lax: Functional Analysis: $2.2$: Index of a Linear Map