Matakuliah
Tahun
: Matrix Algebra for Statistics
: 2009
Projection
Pertemuan 10
Theorm
The orthogonal projection of an mx1 vector x
onto a vector space S can be conveniently
expressed in matrix form.
Let {Z1,Z2, . . . zr,} be any orthonormal basis for S
while {Z1, Z2... ,zm} is an orthonormal basis for Rm
Suppose α1 + α2+ ... + αm are the constants
satisfying the relationship
x= (α1Z1 + α2Z2+ ... + αrZr) + (α1+1Zr+1,+ ... + αmZm)
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Suppose the columns of the mxr
matrices Z1 and W1 each form an
orthononnal basis for the
r-dimensional vector space S.
Then Z1Z1T = W1 W1T
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If mxr matrix X1 = (x1,x2, ... ,xr), the columns
of Z1=X1A will form an orthonormal basis for S if
A is any r x r matrix for which
Z1T Z1=ATX1T X1A=Ir A=nonsingular, Rank (X1)=
Rank (Z1)=r
X1TX1 = (A-1)TA-1 or (X1TX1)-1 = AAT
Consequently, we can obtain an
expression for the projection matrix Ps onto
the vector space S in tellus of X1 as
Ps = Z1T Z1=X1AAT X1T=X1(X1TX1)-1X1T
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Example
Supposed
x1=[1 1 1 1]T, x2=[1 -2 1 -2]T ,
x3=[3 1 1 -1]T
are basis that can be used to form
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and it is easy to verify that
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The projection matrix for the vector space
S spanned by {X1,X2,X3} is given by
The idea can be generalized to the
multiple regression model
y=β0+ β1x1 + . . . + βkxk+ε
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Example
Matrix P is symetric and idempotent
So P is projection matrix
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Definition
An n x n matrix T is orthogonal if T'T = I,. It
immediately follows by taking
determinants that T is nonsingular, T' = T-1
and TT' = I,.
An n x n complex matrix is unitary if U*U =
I, and then U-1 = U*.
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