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Ravindra B. Bapat - Linear Algebra and Linear Models

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Ravindra B. Bapat Linear Algebra and Linear Models
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Linear Algebra and Linear Models comprises a concise and rigorous introduction to linear algebra required for statistics followed by the basic aspects of the theory of linear estimation and hypothesis testing. The emphasis is on the approach using generalized inverses. Topics such as the multivariate normal distribution and distribution of quadratic forms are included.

For this third edition, the material has been reorganised to develop the linear algebra in the first six chapters, to serve as a first course on linear algebra that is especially suitable for students of statistics or for those looking for a matrix theoretic approach to the subject. Other key features include:

coverage of topics such as rank additivity, inequalities for eigenvalues and singular values;

a new chapter on linear mixed models;

over seventy additional problems on rank: the matrix rank is an important and rich topic with connections to many aspects of linear algebra such as generalized inverses, idempotent matrices and partitioned matrices.

This text is aimed primarily at advanced undergraduate and first-year graduate students taking courses in linear algebra, linear models, multivariate analysis and design of experiments. A wealth of exercises, complete with hints and solutions, help to consolidate understanding. Researchers in mathematics and statistics will also find the book a useful source of results and problems.

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Ravindra B. Bapat - Linear Algebra and Linear Models
Linear Algebra and Linear Models
Ravindra B. Bapat
Linear Algebra and Linear Models

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R.B. Bapat Universitext Linear Algebra and Linear Models 3rd ed. 2012 10.1007/978-1-4471-2739-0_1 Springer-Verlag London Limited 2012
1. Vector Spaces and Subspaces
R. B. Bapat 1
(1)
Indian Statistical Institute, New Delhi, India
Abstract
Preliminaries concerning matrices and matrix operations are reviewed. Properties of determinant are recalled without proof. Vector spaces, linear independence, basis and dimension are introduced. It is shown that any two bases of a vector space have the same cardinality and that two vector spaces are isomorphic if and only if they have the same dimension. Extension of a linearly independent set to a basis is considered.
1.1 Preliminaries
In this chapter we first review certain basic concepts. We consider only real matrices. Although our treatment is self-contained, the reader is assumed to be familiar with the basic operations on matrices. We also assume knowledge of elementary properties of the determinant.
An m n matrix consists of mn real numbers arranged in m rows and n columns. The entry in row i and column j of the matrix A is denoted by a ij . An m 1 matrix is called a column vector of order m ; similarly, a 1 n matrix is a row vector of order n . An m n matrix is called a square matrix if m = n .
If A and B are m n matrices, then A + B is defined as the m n matrix with ( i , j )-entry a ij + b ij . If A is a matrix and c is a real number then cA is obtained by multiplying each element of A by c .
If A is m p and B is p n , then their product C = AB is an m n matrix with ( i , j )-entry given by
Linear Algebra and Linear Models - image 1
The following properties hold:
The transpose of the m n matrix A denoted by A is the n m matrix whose i - photo 2
The transpose of the m n matrix A , denoted by A , is the n m matrix whose ( i , j )-entry is a ji . It can be verified that ( A )= A , ( A + B )= A + B and ( AB )= B A .
A good understanding of the definition of matrix multiplication is quite useful. We note some simple facts which are often required. We assume that all products occurring here are defined in the sense that the orders of the matrices make them compatible for multiplication.
(i)
The j -th column of AB is the same as A multiplied by the j -th column of B .
(ii)
The i -th row of AB is the same as the i -th row of A multiplied by B .
(iii)
The ( i , j )-entry of ABC is obtained as
Linear Algebra and Linear Models - image 3
where ( x 1,, x p ) is the i -th row of A and ( y 1,, y q ) is the j -th column of C .
(iv)
If A =[ a 1,, a n ] and
Linear Algebra and Linear Models - image 4
where a i denote columns of A and Linear Algebra and Linear Models - image 5 denote rows of B , then
Linear Algebra and Linear Models - image 6
A diagonal matrix is a square matrix A such that a ij =0, i j . We denote the diagonal matrix
Linear Algebra and Linear Models - image 7
by Linear Algebra and Linear Models - image 8 . When i =1 for all i , this matrix reduces to the identity matrix of order n , which we denote by I n or often simply by I , if the order is clear from the context. Observe that for any square matrix A , we have AI = IA = A .
The entries a 11,, a nn are said to constitute the (main) diagonal entries of A . The trace of A is defined as
Linear Algebra and Linear Models - image 9
It follows from this definition that if A , B are matrices such that both AB and BA are defined, then
Linear Algebra and Linear Models - image 10
The determinant of an n n matrix A , denoted by | A |, is defined as
where the summation is over all permutations 1 n of 1 n and is - photo 11
where the summation is over all permutations {(1),,( n )} of {1,, n } and () is 1 or 1 according as is even or odd.
We state some basic properties of determinant without proof:
(i)
The determinant can be evaluated by expansion along a row or a column. Thus, expanding along the first row,
where A 1 j is the submatrix obtained by deleting the first row and the j -th - photo 12
where A 1 j is the submatrix obtained by deleting the first row and the j -th column of A . We also note that
ii The determinant changes sign if two rows or columns are interchanged - photo 13
(ii)
The determinant changes sign if two rows (or columns) are interchanged.
(iii)
The determinant is unchanged if a constant multiple of one row is added to another row. A similar property is true for columns.
(iv)
The determinant is a linear function of any column (row) when all the other columns (rows) are held fixed.
(v)
| AB |=| A || B |.
The matrix A is upper triangular if a ij =0, i > j . The transpose of an upper triangular matrix is lower triangular .
It will often be necessary to work with matrices in partitioned form. For example, let
be two matrices where each A ij B ij is itself a matrix If compatibility for - photo 14
be two matrices where each A ij , B ij is itself a matrix. If compatibility for matrix multiplication is assumed throughout then we can write
12 Vector Spaces A nonempty set S is called a vector space if it satisfies the - photo 15
1.2 Vector Spaces
A nonempty set S is called a vector space if it satisfies the following conditions:
(i)
For any x , y in S , x + y is defined and is in S . Further,
ii There exists an element in S denoted by 0 such that x 0 x for all x - photo 16
(ii)
There exists an element in S , denoted by 0, such that x +0= x for all x .
(iii)
For any x in S there exists an element y in S such that x + y =0.
(iv)
For any x in S and any real number c , cx is defined and is in S ; moreover, 1 x = x for any x .
(v)
For any x 1, x 2 in S and reals c 1, c 2, c 1( x 1+ x 2)= c 1 x 1+ c 1 x 2, ( c 1+ c 2) x 1= c 1 x 1+ c 2 x 1 and c 1( c 2 x 1)=( c 1 c 2) x 1.
Elements in S are called vectors . If x , y are vectors then the operation of taking their sum x + y is referred to as vector addition. The vector in (ii) is called the zero vector. The operation in (iv) is called scalar multiplication . A vector space may be defined with reference to any field. We have taken the field to be the field of real numbers as this will be sufficient for our purpose.
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