代写STAT3600 Linear Statistical Analysis Chapter 2 Matrix and Distribution Theory代写C/C++语言

DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE

STAT3600 Linear Statistical Analysis

Chapter 2 Matrix and Distribution Theory

2 Matrix

A matrix is a rectangular array of numbers. A is an n × m matrix if it contains n rows and m columns of elements, where ai j represents the element of A at the i-th row and j-th column (i ∈ {1,...,n}, j ∈ {1,...,m}). Mathematically, we denote

When n = 1, A is a 1 × m matrix or a row vector. When m = 1, A is an n × 1 or a column vector. When m = n = 1, A is just a scalar.

2.1 Basic Matrix Operation

1. Matrix Addition:

Let A and B be both n×m matrices. C = A+B is an n×m matrix with elements C =  (cij) where cij = ai j +bij. Similarly, D = A−B is also an n×m matrix with elements D = (dij) where dij = aij − bij.

2. Matrix Multiplication:

Suppose A is an n ×m matrix and B is an m ×p matrix. Then C = AB is an n ×p matrix with elements C = (cij) where

Note that AB ≠ B A. Moreover let k be a scalar, then D = k A is an n × m matrix with element D = (dij), dij = kaij.

2.2 Some Matrix Definition

1. Transpose:

Let A be an n × m matrix, an m × n matrix B = AT (or A') is called the transpose of A with elements B = (bij) where bij = aji.

It is also easy to see that (AT)T = A and (AB)T = BTAT.

2. Square Matrix:

A is a square matrix when n = m.

3. Symmetric:

A is symmetric when A is square and aij = aji for all i and j. For a symmetric matrix, A = AT.

4. Idempotent:

A is idempotent when A2 = A.

5. Identity Matrix:

A is an identity matrix, denoted by In or I , when aii = 1 for all i and aij = 0 for all i ≠ j.

6. Rank:

Let A be an n ×m matrix. The rank of A, denoted by rank(A), is defined as the maximum number of linearly independent column (or row) vectors of A. A is of full ranked if rank(A) = min(n, m). Moreover we have

rank(A) = rank(AT) ≤ min(n, m)

7. Inverse Matrix:

Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I , then B is the inverse of A, and is denoted by B = A−1.

Some properties about the inverse of a matrix are listed below:

• AB = I → B A = I.

• If A−1 exists, then A is called a non-singular or invertible matrix.

• A square matrix A has an inverse if and only if it is of full ranked.

• The inverse is unique if it exists.

• (A-1)-1 = A.

• (AT) −1 = (A−1)T.

• (AB)−1 = B−1 A−1. (Note: A and B are square matrices)

• (kA)−1 = k/1A-1 for some scalar k.

Inversion of 2 × 2 matrices:

Inversion of 3 × 3 matrices:

where det(A) = a (ek − fh) − b(kd − fg) + c(dh − eg).

8. Trace

For an n × n matrix A, the trace is defined as the sum of its diagonal elements

Moreover, we have tr(AB) = tr(B A).

9. Quadratic form.

Let W be a function of n variables Y1 ,...,Yn where

where aij is a given set of numbers. The above form. can be written as W = YTAY where A is an n × n matrix with elements A = (aij), Y = (Y1 ,...,Yn )T is a n × 1 column vector, and W is defined to be a quadratic form. in the n variables Yi .

10. Nonnegative and positive definite

An n × n matrix A is said to be nonnegative definite (or positive semi-definite) if it is symmetric and YTAY ≥ 0 for any n × 1 vector Y .

It is positive definite if YTAY > 0 for any nonzero n × 1 vector Y .







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