Question

If Ymax is the largest eigenvalue of a real symmetric matrix A, show that no diagonal entry of A can be longer than Ymax.

Answer 1

don't think that is true....it doesn't sounds right.....

Answer 2

Maybe I'm reading it wrong but if you have an eigenvalue of 1 in a matrix does that mean that there can not be more than 1 entries for the diagonal.

Answer 3

No, it's saying that if the largest eigenvalue of a matrix is 1, then you cannot have a diagonal entry greater than 1.

Answer 4

Well, it makes sense intuitively. If the matrix A is diagonal then all of the diagonal entries are the eigenvalues so it's trivial. If it isn't diagonal, it's similar to that diagonal matrix, and since it is symmetric, if you tried to increase the value of one of the diagonal entries by adding another row to it, for example, it would be decreased back by the inverse operation. The key point is that the matrix A is symmetric.