Stat701 HW01
1. S = Span(x1 , ., xk ) is a subspace of V where x1 , ., x2 are orthonormal.
(1) Suppose x S . Show that x = x, x1 x1 + + x, xk xk .
Pf: x S x = 1 x1 + + k xk .
But x, xi = 1 x1 + + k xk
Stat 701, HW14
1. Let ei be the ith column of In and D =
n
i=1 (ei
ei e )
i
(1) Show that the columns of D are orthonormal
D Rn
D D
has orthonormal columns if D D = In . But
n
n
= [ i=1 (ei ei e )] j
Stat 701, HW11
1. Prove R(A+ ) = R(A )
: R(A+ ) = R(A (AA )+ ) R(A )
: R(A ) = R(A+ A) R(A+ ).
2. Prove N (A+ ) = N (A )
N (A+ ) = R(I (A+ )+ A+ ) = R(I (A )+ A) = N (A ).
3. Find B such that R (A+ )
Stat 701 HW08
1. Suppose 0 = x Rn and 0 = y Rm .
(1) Find x+ .
x
x
1
x has SVD x =
Thus x = 1 x
+
x 1
x
x
=
x
x2 .
(2) Find (yx )+ .
yx has SVD yx =
Thus (yx )+ =
x
x
y
y
(x y )
(x y )1
x
x
y
y
Stat 701, HW 07
1. Show Theorem 3.2.1 on p112 using max-min formula.
h (A) + m (B ) h (A + B ):
=
=
=
=
h (A) + m (B )
maxcfw_min[ xxAx : 0 = x Rm , x W = 0] W Rm(mh) + m (B )
x
maxcfw_min[ xxAx + m
Stat701 HW06
1. Show that if A Rnn is idempotent, i.e., A2 = A, then the eigenvalues of
A are either 0 or 1.
is an eighen value of A
x = Ax, x = 0
x = Ax = A(Ax) = Ax = 2 x, x = 0
(1 )x = 0, x = 0
(1
Stat701 HW05
1. p129 3.5
xi is an eigenvector of A w.r.t. eigenvalue i , i = 1, ., m
Axi = i xi , xi = 0, i = 1, ., m
(A + I )xi = i xi + xi = (i + )xi , xi = 0 for i = 1, ., m
xi is an eigenvector
UNF Digital Commons
UNF Theses and Dissertations
2006
The Kronecker Product
Bobbi Jo Broxson
University of North Florida
Recommended Citation
Broxson, Bobbi Jo, "The Kronecker Product" (2006). UNF The