# Tag Info

Accepted

### Inverting lower triangular matrix in time $n^2$

No such method is known at present. If one could invert lower triangular $n \times n$ matrices in time $O(n^2)$ then one could multiply $N \times N$ matrices in time $O(N^2)$. Indeed let $n=3N$ and ...
• 77.5k
Accepted

• 4,979

### Should the formula for the inverse of a 2x2 matrix be obvious?

For $\mathbf{A}$ near zero we have $$(1-\mathbf{A})^{-1}\approx 1+\mathbf{A}$$ so it has to negate the off-diagonals. (If you want to get all fancy about it you could notice that we use $\exp$ to ...
• 8,116

• 178k
Accepted

Accepted

### If $\Vert A-B\Vert_{op}\leq \varepsilon$ then $A^{-1}$ and $B^{-1}$ are uniformly close

$A^{-1}-B^{-1}=B^{-1}(B-A)A^{-1}$ so $\|A^{-1}-B^{-1}\| \le \delta_0^{-2} \epsilon$. This is the best possible bound in terms of the given parameters, as you can see by considering 2 by 2 diagonal ...
• 14k
Accepted

### Full-rank matrix

OK, let's call the block matrix above $M$. First eliminate $N$ by a substitution $c\mapsto d N$. Then substitute $z_i \mapsto d y_i$ to eliminate $d$. Then you can construct the Schur complement w.r.t....
• 2,705
Accepted

Accepted

• 77.5k