## How to Find Moore Penrose Inverse

I have a matrix: A= \begin{bmatrix} -1 & 0 & 1 & 2 \\ -1 & 1 & 0 & -1 \\ 0 & -1 & 1 & 3 \\ 0 & 1 & -1 & -3 \\ 1 & -1 & 0 & 1 \\ 1 & 0 & -1 & -2 \\ \end{bmatrix} … Read more

## Inverse of the Jordan block matrix

There is the Jordan block matrix Jλ(n):=(λ1λ1……λ1λ)∈Cn×n How to find the inverse of this matrix? I tried with the Gauss Jordan Elimination and got Jλ(n)−1=(1λ01λ0……1λ01λ) But i don’t know if this works. Answer Your matrix Jλ(n)=λI+N where N=(010⋯00001⋯00⋮⋮⋮⋱⋮⋮000⋯10000⋯01000⋯00). Then N is nilpotent: Nn=0 and so I+tN will have the inverse I−tN+t2N2−⋯+(−t)n−1Nn−1. Then Jλ(n)−1=λ−1(1+λ−1N)−1=λ−1(I−λ−1N+λ−2N2−⋯+(−λ)−n+1Nn−1). AttributionSource : … Read more

## Inverse by left multiplication but not right?

Suppose T:V→W and U:W→V are linear transformations. It is known that U=T−1 if UT=IV and TU=IW. Is it possible to then also have a transform Z:W→V such that ZT=IV but TZ≠IW (and likewise, TZ=IW but ZT≠IV)? Answer It is indeed possible. For instance, take W=R2,V=R, Z(x,y)=x If we define T(x)=(x,0), then we indeed have ZT=IV … Read more