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 , xi = i .
Hence x = x, x1 x1 + + x, xk xk .
(2) Suppo
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 =1 (ej ej e )
j
i
2
n
=
=
=
=
n
n
j =1 (ei
i=1
n
e ei
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
=
xy
x2 y 2 .
2. Express A , AgR , AgL , AgLR and A+
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 (B ) : 0 = x Rm , x W = 0] W Rm(mh)
x
maxcfw_min[ xxAx
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 ) = 0
= 0 or = 1.
(
)
12
2. A =
21
(1) Find orthogona
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 of A + I w.r.t. eigenvalue i + , for i = 1, ., m.
2. p
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 Theses and Dissertations. Paper 25.
http:/digitalcommons.u