Largest singular value norm of a matrix

For a m times n matrix , we define the largest singular value (or, LSV) norm of to be the quantity

This quantity satisfies the conditions to be a norm (see here). The reason for which this norm is called this way is given here.

The LSV norm can be computed as follows. Let us square the above. We obtain a representation of the squared LSV norm as a Rayleigh quotient of the matrix A^TA :

$|A|^2 = max_{x ::: |x|_2 = 1} : x^T A^TA x.$

This shows that the squared LSV norm is the largest eigenvalue of the (positive semi-definite) symmetric matrix A^TA , which is denoted $lambda_{rm max}$ . That is:

$|A| = sqrt{lambda_{rm max} (A^TA)}.$