NLA MT25, Useful miscellany
Flashcards
What is
\[\begin{bmatrix}
I _ m & X \\ 0 & I _ n
\end{bmatrix}^{-1}\]
?
\[\begin{bmatrix}
I _ m & -X \\ 0 & I _ n
\end{bmatrix}\]
Suppose that $ \vert \vert X \vert \vert < 1$ in a submultiplicative norm. How can you rewrite $(I - X)^{-1}$?
\[(I - X)^{-1} = I + X + X^2 + X^3 + \cdots\]
(submultiplicativity is useful so you can say $ \vert \vert X^k \vert \vert \le \vert \vert X \vert \vert ^k \to 0$).
Suppose that $Q \in \mathbb R^{m \times n}$ is orthogonal (note that $Q$ might be rectangular!). What does this mean, and what does this not mean?
\[Q^\top Q = I _ n\]
This does not mean that
\[Q Q^\top = I _ n\](although this is true for square matrices, and more generally $A^{-1} A = I _ n$ implies $A A^{-1} = I _ n$, it is not true in general for rectangular matrices).