I am studying $LU$ decomposition with partial pivoting in Numerical Linear Algebra book. I have a problem in understanding the discussion on $L^{-1}$ in lecture 22.(Original paragraph is attached as an image.enter image description here

In $LU$ decomposition(with partial pivoting) of a $n \times n$ matrix $A$, we have $LU = PA$.

$P$ is a $n \times n$ permutation matrix.

$L$ and $U$ are $n \times n$ lower and upper triangular matrices, respectively.

We know the column spaces of $PA$ and $L$ are the same. Now it is discussed:

if $A$ is a random matrix, the column spaces of the matrix are oriented randomly, and the same is true for column spaces of $P^{-1}L$. However this condition is incompatible with $L^{-1}$ being large. It can be shown that if $L^{-1}$ is large, then the column spaces of $L$, or any permutation $P^{-1}L$, must be skewed in a fashion that is very far from random.

The book discussion is supported by few sparsity plots. My question is how can we show when $L^{-1}$ is large, the column spaces of the matrix are not oriented randomly?

Thank you for any help.


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