Advanced Linear Algebra Convexity Exercises Help

1. Full correct solutions.

2. Please, if you’re not sure you can solve these questions, don’t bid.

3. Preferable solutions from the lectures material/theorems, but it’s not a requirement. any correct solution will do.

Side note: the material is based on the book “Linear Algebra in Action”. (You can check there ‘kinda’ similar questions and explanations)

(1) In class we showed that if A € Cnxn and A > 0, then
(det A)1/n = min
trace AC
:Ce Chx, C> 0 and det C =
=1}
n
(1)
(a) Show that the formula (1) can be reexpressed as
-1}
п
traceCÁC
(det A)1/n = min
: Cecnxn, C> 0 and det C =
(2)
(b) Show by example that the formula (1) is false if A is singular.
[REMARK: The point in (b) is to show that there does not exist a
matrix C in the indicated class that achieves the minimum.]
(2) Showed that if A e Cnxn, A 0 and A is singular, then there exists
a sequence of matrices Ck, k = 1,2,…, such that Ck0, det k = 1
and trace ACk < 1/k. (3) Show by direct estimates that if AER***, A > 0 and b € R” then the
function
3 (Ax,x) – (b, x) with x
is convex on R.
(4) Let p(r) belong to the class C²(a,b) of real valued functions that have
continuous first and second derivatives the interval (a,b). Show that
(a) If a < a < x. then 4(x) = f(a) + (x - a)'(a) + (x - 5)"(s)ds for a