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

    \begin{align*} ||A|| := \max\limits_{x} ||Ax||_2: ||x||=1. \end{align*}

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:

    \begin{align*} ||A||^2 = \max\limits_{x: ||x||=1} x^TA^TAx. \end{align*}

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:

    \begin{align*} ||A|| = \sqrt{\lambda_{\max} (A^TA)}. \end{align*}

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