Some Results on Linear Algebra, Matrix Theory and Distributions

Chapter 1
Some Results on Linear Algebra, Matrix Theory and Distributions


We need some basic knowledge to understand the topics in the analysis of variance.


Vectors:
A vector Y is an ordered n-tuple of real numbers. A vector can be expressed as a row vector or a column
vector as
 y1 
 
y
Y  2
 
 
 yn 
is a column vector of order n  1 and
Y '  ( y1 , y2 ,..., yn )
is a row vector of order 1 n.


If all yi  0 for all i = 1,2,…,n then Y '  (0, 0,..., 0) is called the null vector.


If
 x1   y1   z1 
     
x y z
X   2 , Y   2 , Z   2 
     
     
 xn   yn   zn 
then
 x1  y2   ky1 
   
 x2  y2   ky2 
X Y  , kY 
   
   
 xn  yn   kyn 


X  (Y  Z )  ( X  Y )  Z
X '(Y  Z )  X ' Y  X ' Z
k ( X ' Y )  ( kX ) ' Y  X '( kY )
k ( X  Y )  kX  kY
X ' Y  x1 y1  x2 y2  ...  xn yn

where k is a scalar.


Analysis of Variance | Chapter 1 | Linear Algebra, Matrix Theory and Dist. | Shalabh, IIT Kanpur
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,Orthogonal vectors:
Two vectors X and Y are said to be orthogonal if X ' Y  Y ' X  0 .


The null vector is orthogonal to every vector X and is the only such vector.


Linear combination:
If x1 , x2 ,..., xm are m vectors and k1 , k2 ,..., km are m scalars, then
m
t   ki xi
i 1


is called the linear combination of x1 , x2 ,..., xm .


Linear independence
If X1 , X 2 ,..., X m are m vectors then they are said to be linearly independent if there exist scalars

k1 , k2 ,..., km such that
m

k X
i 1
i i  0  ki  0 for all i = 1,2,…,m.

m
If there exist k1 , k2 ,..., km with at least one ki to be nonzero, such that k x
i 1
i i  0 then

x1 , x2 ,..., xm are said to be linearly dependent.

 Any set of vectors containing the null vector is linearly dependent.
 Any set of non-null pair-wise orthogonal vectors is linearly independent.
 If m > 1 vectors are linearly dependent, it is always possible to express at least one of them as a
linear combination of the others.


Linear function:
Let K  (k1 , k2 ,..., km ) ' be a m  1 vector of scalars and X  ( x1 , x2 ,..., xm ) be a m  1 vector of variables,
m
then K ' Y   ki yi is called a linear function or linear form. The vector K is called the coefficient
i 1


vector. For example, the mean of x1 , x2 ,..., xm can be expressed as

 x1 
 
1 m 1 x 1
x   xi  (1,1,...,1)  2   1'm X
m i 1 m   m
 
 xm 
where 1'm is a m  1 vector of all elements unity.
Analysis of Variance | Chapter 1 | Linear Algebra, Matrix Theory and Dist. | Shalabh, IIT Kanpur
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, Contrast:
m m
The linear function K ' X   ki xi is called a contrast in x1 , x2 ,..., xm if k i  0.
i 1 i 1


For example, the linear functions
x1 x
x1  x2 , 2 x1  3x2  x3 ,  x2  3
2 3
are contrasts.
m
 A linear function K ' X is a contrast if and only if it is orthogonal to a linear function x
i
i or to

1 m
the linear function x   x.i .
m i 1
 Contrasts x1  x2 , x1  x3 ,..., x1  x j are linearly independent for all j  2, 3,..., m.

 Every contrast in x1 , x2 ,..., xn can be written as a linear combination of (m - 1) contrasts

x1  x2 , x1  x3 ,..., x1  xm .


Matrix:
A matrix is a rectangular array of real numbers. For example
 a11 a12 ... a1n 
 
 a21 a22 ... a2 n 
   
 
 am1 am 2 ... amn 

is a matrix of order m  n with m rows and n columns.
 If m = n, then A is called a square matrix.
 If aij  0, i  j , m  n, then A is a diagonal matrix and is denoted as

A  diag (a11 , a22 ,..., amm ).

 If m = n (square matrix) and aij  0 for i > j , then A is called an upper triangular matrix. On

the other hand if m = n and aij  0 for i < j then A is called a lower triangular matrix.

 If A is a m  n matrix, then the matrix obtained by writing the rows of A and
columns of A as columns of A and rows of A respectively is called the transpose of a matrix A
and is denoted as A ' .
 If A  A ' then A is a symmetric matrix.
 If A   A ' then A is skew-symmetric matrix.
 A matrix whose all elements are equal to zero is called a null matrix.


Analysis of Variance | Chapter 1 | Linear Algebra, Matrix Theory and Dist. | Shalabh, IIT Kanpur
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