Let a be real symmetric and positive semidefinite. let α1 ≤ · · · ≤ αn be the eigenvalues of a. show that for any nonzero vector x ∈ rn, we have α1 ≤ ila xll / xll ≤ αn.

Consider an example of a deck of 52 cards: example set of 52 playing cards: 13 of each suit clubs, diamonds, hearts, and spades ace 2 3 4 5 6 7 8 9 10 jack queen king clubs diamonds hearts spades what is the probability of drawing three queens from a standard deck of cards, given that the first card drawn was a queen? assume that the cards are not replaced.