Proving a Theorem, related to Gerschgorin

[syj]

Homework Statement

Here is the theorem I need to prove:

For A=(a_ij)\inC_nxn

we have

p(A)\leqmax_{i}\Sigma^{n}_{j=1}|a_ij|

Homework Equations

The Attempt at a Solution

I have no idea how to go about this.

 

[CompuChip]

Some definitions might be nice.
What is C_{n \times n}? What is p(A)? What does i run over?

 

[syj]

Sorry,
C_nxn is the set of nxn matrices with entries in the complex number system.

p(A)= max{|\lambda₁|, |\lambda₂|, ...}

p(A) is the smallest circle in the comple plane, centered at the origin, which contains all the characteristic values of A.

 

[CompuChip]

Maybe it's a good idea to start with the simple case, where A is diagonalizable.