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### 21.4 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 one way to normalize columns of a matrix X to unit norm:

      s = norm (X, "columns");
      X /= diag (s);

The same can also be accomplished with broadcasting (see Broadcasting):

      s = norm (X, "columns");
      X ./= s;

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 to 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;