Largest singular value norm of a matrix

L. El Ghaoui

Largest singular value norm of a matrix

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

$||A|| := \max\limits_{x} ||Ax||_2: ||x||=1.$

This quantity satisfies the conditions to be a norm (see here). The reason why 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 $A^TA$ :

$||A||^2 = \max\limits_{x: ||x||=1} x^TA^TAx.$

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

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

Largest singular value norm of a matrix

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