I accidently found a class of rank-deficient square matrices that are their own Moore-Penrose pseudo inverse:


$$\boldsymbol{A}\in\mathbb{R}^{n\times n}: a\_{(i,i)}=n-1,\, a\_{\lbrace i,j\rbrace}=-1\implies\Big(\frac{1}{n}\boldsymbol{A}\Big)^+ =\Big(\frac{1}{n}\boldsymbol{A}\Big)$$


as an exemplary $4\times 4$ calculation done with WA demonstrates


![](https://i.stack.imgur.com/8i1yO.png)

> 
> **Questions:**
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> * have all Moore-Penrose "pseudo unit-matrices" been characterized, what about other kinds of pseudo inverses?
> * do these "pseudo unit-matrices" play a special role in the theory of pseudo inverses, resp., in linear algebra?
> 
> 
> 



Characterization of pseudo unit-matrices
All symmetric idempotent matrices are their own Moore-Penrose pseudo-inverse; these are matrices with eigenvalues equal to either 0 or 1.


Your matrices are in this class.


