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### 21.4 Some Examples of Usage

The following can be used to solve a linear system `A*x = b` using the pivoted LU factorization:

```       [L, U, P] = lu (A); ## now L*U = P*A
x = U \ L \ P*b;
```

This is how you normalize columns of a matrix X to unit norm:

```       s = norm (X, "columns");
X = diag (s) \ X;
```

The following expression is a way to efficiently calculate the sign of a permutation, given by a permutation vector p. It will also work in earlier versions of Octave, but slowly.

```       det (eye (length (p))(p, :))
```

Finally, here's how you solve a linear system `A*x = b` with Tikhonov regularization (ridge regression) using SVD (a skeleton only):

```       m = rows (A); n = columns (A);
[U, S, V] = svd (A);
## determine the regularization factor alpha
## alpha = ...
## transform to orthogonal basis
b = U'*b;
## Use the standard formula, replacing A with S.
## S is diagonal, so the following will be very fast and accurate.
x = (S'*S + alpha^2 * eye (n)) \ (S' * b);
## transform to solution basis
x = V*x;
```