## Saturday, December 18, 2010

### old friends keep you honest.

it's good to have people who know you when you were young. a friend from my undergraduate years, she's begun a postdoc in the same department as me. tonight she reminded me of two things:

1. at some point i once arrived early at a particular TA meeting. the main instructor was late and apparently in the meanwhile time i started ranting (happily) about gershgorιn's circle theοrem for .. 5 minutes?
Let $A$ be a complex $n \times n$ matrix, with entries $a_{ij}$. For $i \in \{ 1, \ldots, n \}$ let $R_i = \sum_{j \neq i} \left|a_{ij}\right|$ be the sum of the absolute values of the non-diagonal entries in the $i$th row. Let $D(a_{ii}, R_i)$ be the closed disc centered at $a_{ii}$ with radius $R_i$. Such a disc is called a Gershgorin disc.

Theorem: Every eigenvalue of $A$ lies within at least one of the Gershgorin discs $D(a_{ii}, R_i)$.

it wasn't until she reminded me that i remembered my love for this theorem.

isn't it cool, though?!? it states that in some cases, it suffices to draw a picture in order to determine the invertιbility of a matrix!

2. to my discredit, though, she told me that i was the first person to tell her about the joke regarding $e^x$ and the differential operator.

that only adds to my infamy .. \-: