# Thread: 2-Norm of PD Matrix

1. ## 2-Norm of PD Matrix

Hi all,

In most books i've seen, it says that:

$\displaystyle ||A||_2 = \textnormal{ Root of largest eigenvalue of } A^T A$

If $\displaystyle A$ is symmetric and positive definite, this just becomes
equal to the largest eigenvalue. However, my teacher says that this even hold for positive semidefinite. Can anybody tell me if they agree/disagree, and also why?

2. Originally Posted by leoemil
Hi all,

In most books i've seen, it says that:

$\displaystyle ||A||_2 = \textnormal{ Root of largest eigenvalue of } A^T A$

If $\displaystyle A$ is symmetric and positive definite, this just becomes
equal to the largest eigenvalue. However, my teacher says that this even hold for positive semidefinite. Can anybody tell me if they agree/disagree, and also why?
Teacher is correct, the result still holds in the positive semidefinite case.

One way to see it, thinking of A as a linear transformation rather than a matrix, is to let L be the kernel of A (equivalently, the eigenspace corresponding to the eigenvalue 0). Let B denote the restriction of A to the orthogonal complement $\displaystyle L^\perp$. Then B is positive definite on $\displaystyle L^\perp$, and $\displaystyle B^{\textsc t}B$ is the restriction of $\displaystyle A^{\textsc t}A$ to $\displaystyle L^\perp$. So the nonzero eigenvalues of $\displaystyle A^{\textsc t}A$ are exactly the eigenvalues of $\displaystyle B^{\textsc t}B$. The square root of the largest one is equal to the largest eigenvalue of B, which is equal to the largest eigenvalue of A.