324 CHAPTER 12. SELF ADJOINT OPERATORS

25. ↑Show that if A is a symmetric positive definite n× n real matrix, then A has an LUfactorization with the property that each entry on the main diagonal in U is positive.Hint: This is pretty clear if A is 1×1. Assume true for (n− 1)× (n− 1). Then

A =

(Â a

aT ann

)

Then as above, Â is positive definite. Thus it has an LU factorization with all positiveentries on the diagonal of U . Notice that, using block multiplication,

A =

(LU a

aT ann

)=

(L 0

0 1

)(U L−1a

aT ann

)

Now consider that matrix on the right. Argue that it is of the form L̃Ũ where Ũhas all positive diagonal entries except possibly for the one in the nth row and nth

column. Now explain why det (A) > 0 and argue that in fact all diagonal entries of Ũare positive.

26. ↑Let A be a real symmetric n× n matrix and A = LU where L has all ones down thediagonal and U has all positive entries down the main diagonal. Show that A = LDHwhere L is lower triangular and H is upper triangular, each having all ones down thediagonal and D a diagonal matrix having all positive entries down the main diagonal.In fact, these are the diagonal entries of U .

27. ↑Show that if L,L1 are lower triangular with ones down the main diagonal and H,H1

are upper triangular with all ones down the main diagonal and D, D1 are diagonalmatrices having all positive diagonal entries, and if LDH = L1D1H1, then L =L1, H = H1, D = D1. Hint: Explain why D−1

1 L−11 LD = H1H

−1. Then explainwhy the right side is upper triangular and the left side is lower triangular. Concludethese are both diagonal matrices. However, there are all ones down the diagonal inthe expression on the right. Hence H = H1. Do something similar to conclude thatL = L1 and then that D = D1.

28. ↑Show that if A is a symmetric real matrix such that x → ⟨Ax,x⟩ is positive definite,then there exists a lower triangular matrix L having all positive entries down thediagonal such that A = LLT . Hint: From the above, A = LDH where L,H arerespectively lower and upper triangular having all ones down the diagonal and D is adiagonal matrix having all positive entries. Then argue from the above problem andsymmetry of A that H = LT . Now modify L by making it equal to LD1/2. This iscalled the Cholesky factorization.

29. Given F ∈ L (X,Y ) where X,Y are inner product spaces and dim (X) = n ≤ m =dim (Y ) , there exists R,U such that U is nonnegative and Hermitian and R∗R = Isuch that F = RU. Show that U is actually unique and that R is determined onU (X) .