Matakuliah
Tahun
: Matrix Algebra for Statistics
: 2009
Matriks Operator
Pertemuan 9
Partitioned Matrices
Purpose: To obtain expressions for the
inverse and determinant of an m x m
matrix A that is partitioned into the 2 x 2
block
where A11 is m1xm1. A12 is m1xm2. A21 is
m2xml, and A22 is m2xm2
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Contoh:
Let Amxm and Bmxm partitioned as
A, A11, and A22 are nonsingular matrices
B = A-1 and partition B as
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Then,
a) B11 =(A11-A12A22-1A21)-1
=A11 +A11A12B22A21A11,
b) B22 = (A22 – A21-1A12)-1
= A22-1 + A22-1 A21B11A12A22-1
c) B12 = -A11-1 A12B22,
d) B21 =-A22-1A21 B11
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Matrix equation
Yields,
A11B11 +A12B21 = Im1
A21B12 +A22B22 = 1m2
A11B12 +A12B22 = (0)
A21 B11 + A22B21 = (0)
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Example 1
Consider the regression model y = Xᵦ + є,
where y is Nx1, X is Nx(k + 1), β is (k + 1)
x1, and є is N x 1.
Suppose that β and X are partitioned as
β= (β 1T, β 2T)T and X = (X1, X2) so that
the product X1 β1is defined
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Example 2
Let the m x m matrix A be partitioned
If A22 = Im2 and A12 = (0) or A21 = (0),
then IAI = IA11I·
To find the determinant
use the cofactor expansion formula for a
determinant
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where B is the (m2 - 1) x m1 matrix
obtained by deleting the last row from A21
Repeating this process another (m2 - 1)
times yields IA I = IA11l.
In a similar way we obtain IA I = IA11I.
when A21 = (0). by repeatedly expanding
along the last row.
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Nonnegatif Vector
An m x n matrix A is a nonnegative
matrix, indicated by A ≥ (0), if each element
of A is nonnegative. Similarly, A
is a positive matrix, indicated by A > (0), if
each element of A is positive.
We will write A ≥ B and A> B to mean that
A-B ≥ (0) and A-B > (0), respectively.
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• Any matrix A can be transformed to a
nonnegative matrix by replacing each
of its elements by its absolute value.
This will be denoted by abs(A); that is,
if A is an mxn matrix, then
abs(A) is also an mxn matrix with (i,j)th
element given by Ia ij I.
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Let A be an mxm matrix and x be an mx1 vector. If
A ≥(0) and x > 0, then
with similar inequalities holding when minimizing
and maximizing over columns instead of rows
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Theorm 1
Let A be an m x m positive matrix. Then
ρ(A) is positive
and is an eigenvalue of A. In addition,
there exists a positive eigenvector of A
corresponding to the eigenvalue ρ(A).
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g
Theorm 2
Let A be an mxm positive matrix and
suppose that  is an eigenvalue of A
satisfying
= ρ(A).
If x is any eigenvector corresponding
to , then A abs(x) = ρ(A)abs(x)
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Theorm 3
If A is an mxm positive matrix,
then the dimension of the eigenspace
corresponding to the eigenvalue ρ(A) is one.
Further, if A is an eigenvalue of A and
Aρ(A), then II < ρ(A).
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VEC OPERATOR
There are situations in which it is useful to
transform a matrix to a vector that has as
its elements the elements of the matrix.
One such situation in statistics involves the
study of the distribution of the sample
covariance matrix S
The operator that transforms a matrix to a
vector is known as the vec operator
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If the mxn matrix A has ai as its i th
column, then vec(A) is the mnx1
vector given by
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Example
A is 2x3 matrix,
If a is mx1 and b is nx1, then abT is mxn and
vec(abT) = vec([b1a, b2a, ... , bna])
=
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Theorm
Let a and b be any two vectors, while A and
B are two matrices of the same size.
Then
a) vec(a) = vec(aT) = a,
b) vec(αbT) = b  a,
c) vec( αA + (βB) = α vec(A) + β vec(B),
where α and β are scalars.
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Example
Suppose that we are interested in the distribution of
the sample covariance matrix or the distribution of
the sample correlation matrix computed from a
sample of observations on three different variables.
The resulting sample covariance and correlation
matrices would be of the form
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So that
vec(S) = (S11 S12, S13, S12, S22, S23, S13, S23, S33)T,
vec(R) = (1, rl2, r13, r12, 1, r23, r13, r23, 1)T
Since both S and R are symmetric, there are
redundant elements in vec(S) and
vec(R). The elimination of these results in v(S) and
v(R) given by
v(S) = (S11, S12, S13, S22, S23, S33)T,
v(R) = (I, rl2, r13, 1, r23, 1)T
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Eliminating the nonrandom 1 s from v(R), we
obtain
which contains all of the random variables in
R.
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