Question

Help please!

Let A be an n×n matrix, whose entries could be real or complex numbers. Suppose
that the diagonal entries of A are much larger than the size of the other entries in the
same row, in the sense that for each row i

2||aii|| > ||ai1|| + ||ai2|| + ||ai3|| + · · · + ||ain||.
Prove that A is invertible. (Hint: How is A not being invertible connected to the
eigenvalues of A?)

Answer 1

All you need to show that the matrix is invertible is to show that the determinant isn't 0. If the diagonal entries of A are much larger, you can make a heuristic argument that the absolute value of the determinant is very large, thus it's invertible.

Answer 2

thank you so much!!

Answer 3

can i ask if you understand this at all?!
The purpose of this question is to figure out what “sufficiently random” means, and why
it is there. We’ll use the matrix
A =(1 −2 2)
(−1 0 2)
(−1 −3 5)

.
(a) Starting with the vector ~w1 = (1, 0, 0), do this procedure 10 times (you don’t need
to show all the calculations, just show the first two and the final answer. You
should re-scale so that the last coordinate is 1).

Answer 4

I actually don't understand that at all... sorry :(

Answer 5

ah its ok! thanks anyways!