Textbook: “Matrix algebra useful for statistics”, Searle
Textbook: “Matrix algebra useful for statistics”, Searle.
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Math Algebra ( Word , PDF )
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• Online Grade: 2008, Summer, Basic Statistics
Objective: introduce basic concepts and skills in matrix algebra. In addition, some applications of matrix algebra in statistics are described.
Section 1. Introduction and Matrix Operations
Definition of [pic] matrix:
An [pic] matrix A is a rectangular array of rc real numbers arranged in r horizontal rows and c vertical columns:
[pic].
The i’th row of A is
[pic],
and the j’th column of A is
[pic]
We often write A as
[pic].
Matrix addition:
Let
[pic],
[pic],
[pic].
Then,
[pic],
[pic]
and the transpose of A is denoted as
[pic]
Example 1:
Let
[pic] and [pic].
Then,
[pic],
[pic]
and
[pic].
Matrix multiplication:
We first define the dot product or inner product of n-vectors.
Definition of dot product:
The dot product or inner product of the n-vectors
[pic] and [pic],
are
[pic].
Example 1:
Let [pic] and [pic]. Then, [pic].
Definition of matrix multiplication:
[pic]
[pic]
[pic]
That is,
[pic]
Example 2:
[pic].
Then,
[pic]
since
[pic], [pic]
[pic], [pic]
[pic], [pic].
Example 3
[pic]
Another expression of matrix multiplication:
[pic]
where [pic] are [pic] matrices.
Example 2 (continue):
[pic]
Note:
Heuristically, the matrices A and B, [pic] and [pic], can be thought as [pic] and [pic] vectors. Thus,
[pic]
can be thought as the multiplication of [pic] and [pic] vectors. Similarly,
[pic]
can be thought as the multiplication of [pic] and [pic] vectors.
Note:
I. [pic] is not necessarily equal to [pic]. For instance, [pic]
[pic].
II. [pic] might be not equal to [pic]. For instance,
[pic]
[pic]
III. [pic], it is not necessary that [pic] or [pic]. For instance,
[pic]
[pic]
IV. [pic], [pic], [pic]
p factors
Also, [pic] is not necessarily equal to [pic].
V. [pic].
Trace:
Definition of the trace of a matrix:
The sum of the diagonal elements of a [pic] square matrix is called the trace of the matrix, written [pic], i.e., for
[pic],
[pic].
Example 4:
Let [pic]. Then, [pic].
Homework 1
1. Prove [pic], where A and B are [pic] and [pic]
matrices, respectively.
2.
(a) When does [pic]
(b) When [pic] Prove [pic]
(c) When [pic], prove [pic]
Section 2 Special Matrices
2.1 Symmetric Matrices:
Definition of symmetric matrix:
A [pic] matrix [pic] is defined as symmetric if [pic]. That is,
[pic].
Example 1:
[pic] is symmetric since [pic].
Example 2:
Let [pic] be random variables. Then,
[pic] [pic] … [pic]
[pic]
is called the covariance matrix, where [pic], is the covariance of the random variables [pic] and [pic] and [pic] is the variance of [pic]. V is a symmetric matrix. The correlation matrix for [pic] is defined as
[pic] [pic] … [pic]
[pic]
where [pic], is the correlation of [pic] and [pic]. R is also a symmetric matrix. For instance, let [pic] be the random variable represent the sale amount of some product and [pic] be the random variable represent the cost spent on advertisement. Suppose
[pic]
Then,
[pic]
and
[pic]
Example 3:
Let [pic] be a [pic] matrix. Then, both [pic] and [pic] are symmetric since
[pic] and [pic].
[pic] is a [pic] symmetric matrix while [pic] is a [pic] symmetric matrix.
[pic]
Also,
[pic]
Similarly,
[pic]
and
[pic]
For instance, let
[pic] and [pic].
Then,
[pic]
In addition,
[pic]
Note:
A and B are symmetric matrices. Then, AB is not necessarily equal to [pic]. That is, AB might not be a symmetric matrix.
Example 4:
[pic] and [pic].
Then,
[pic]
Properties of [pic] and [pic]:
(a)
[pic]
(b)
[pic]
[proof]
(a)
Let
[pic]
[pic].
Thus, for [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
(b)
Since [pic]
[pic]
By (a),
[pic]
Note:
A [pic] matrix [pic] is defined as skew-symmetric if [pic]. That is,
[pic].
Example 5:
[pic]
Thus,
[pic][pic].
2.2 Idempotent Matrices:
Definition of idempotent matrices:
A square matrix K is said to be idempotent if
[pic]
Properties of idempotent matrices:
1. [pic] for r being a positive integer.
2. [pic] is idempotent.
3. If [pic] and [pic] are idempotent matrices and [pic]. Then,
[pic] is idempotent.
[proof:]
1.
For [pic] [pic].
Suppose [pic]is true, then [pic].
By induction, [pic] for r being any positive integer.
2.
[pic]
3.
[pic]
Example 1
Let [pic] be a [pic] matrix. Then,
[pic] is an idempotent matrix since
[pic].
Note:
A matrix A satisfying [pic] is called nilpotent, and that for which [pic] could be called unipotent.
Example 2:
[pic] [pic] A is nilpotent.
[pic] [pic] B is unipotent.
Note:
[pic] is a idempotent matrix. Then, [pic] might not be idempotent.
2.3 Orthogonal Matrices:
Definition of orthogonality:
Two [pic] vectors u and v are said to be orthogonal if
[pic]
A set of [pic] vectors [pic] is said to be orthonormal if
[pic]
Definition of orthogonal matrix:
A [pic] square matrix P is said to be orthogonal if
[pic].
Note:
[pic]
[pic] [pic]
Thus,
[pic] and [pic]
are both orthonormal sets!!
Example 1:
(a) Helmert Matrices:
The Helmert matrix of order n has the first row
[pic],
and the other n-1 rows ([pic]) has the form,
[pic]
(i-1) items n-i items
For example, as [pic], then
[pic]
In statistics, we can use H to find a set of uncorrelated random variables. Suppose [pic] are random variables with
[pic]
Let
[pic]
Then,
[pic]
since [pic] is an orthonormal set of vectors. That is, [pic] are uncorrelated random variables. Also,
[pic],
where
[pic].
(b) Givens Matrices:
Let the orthogonal matrix be
[pic]
G is referred to as a Givens matrix of order 2. For a Givens matrix of order 3, there are [pic] different forms,
1 2 3 1 2 3
[pic].
The general form of a Givens matrix [pic] of order 3 is an identity matrix except for 4 elements, [pic] and [pic] are in the i’th and j’th rows and columns. Similarly, For a Givens matrix of order 4, there are [pic] different forms,
1 2 3 4 1 2 3 4
[pic]
1 2 3 4 1 2 3 4
[pic]
1 2 3 4 1 2 3 4
[pic].
For the Givens matrix of order n, here are [pic] different forms. The general form of [pic] is an identity matrix except for 4 elements,
[pic].
2.4 Positive Definite Matrices:
Definition of positive definite matrix:
A symmetric [pic] matrix A satisfying
[pic] for all [pic],
is referred to as a positive definite (p.d.) matrix.
Intuition:
If [pic] for all real numbers x, [pic], then the real number a is positive. Similarly, as x is a [pic] vector, A is a [pic] matrix and [pic], then the matrix A is “positive”.
Note:
A symmetric [pic] matrix A satisfying
[pic] for all [pic],
is referred to as a positive semidefinite (p.d.) matrix.
Example 1:
Let
[pic] and [pic].
Thus,
[pic]
Let [pic]. Then, A is positive semidefinite since for [pic]
[pic].
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