Validity condition for time-independent perturbation theory

[cks]

H=H0 + lambda * W

lambda << 1 must hold and the matrix elements of W are comparable in magnitude to those of H0.

More precisely, the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0.

I don't understand what is the meaning of " the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0".

(the above explanation are obtained from the SChaum's Outlines of Quantum Mechanics)

 

[cks]

the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0".

Let's say the matrix W=[2.2 3.1 4.1; 4.1 5.3 6.0; 7.3 8.2 9.3] (matlab code)

let's say the eigenvalues of H0 are 1 2 3 4 5 6 7 8 9

the matrix element 2.2 is roughly the same as the difference of the eigenvalues of 3-1. Am I understanding this correctly?

the matrix elements of W are of the "same magnitude"(don't understand what same magnitude means?) as the difference(difference? difference between which eigenvalues, in my example, there are 9 eigenvalues, which minus which is the difference the author is talking?) between the eigenvalues of H0".