1.2 Vector

A matrix with column order is called a column vector \begin {equation} \tag {2} \underline {a} = \begin {pmatrix} a_1\\ a_2\\ \vdots \\ \vdots \\ a_n\\ \end {pmatrix} \end {equation} its transponse \(\underline {a}'= \begin {pmatrix} a_1, & a_2, \ldots , & a_n\\ \end {pmatrix}\) is a row vector.

Again

1.: If the entries are constants then the vector is constant vector.
2.: If the entries are random variables then the vector is a random vector.

It is possible to write a matrix in equation (1) as vector.
i.e \(A = \begin {pmatrix} \underline {a}_1, & \underline {a}_2, \ldots , & \underline {a}_p\\ \end {pmatrix}\) , where \(\underline {a}_i\) is the \(i^{\text {th}}\) column vector.

Example 1.2. Data on Incidence of poverty by province from 1991, 1993 and 1996.


Province	1991	1993	1996

Central	55.7	70.7	58.6
Copperbelt	43.8	28.1	33.5
Eastern	76.1	81.2	69.9
Luapula	72.5	77.8	63.9
Lusaka	18.7	24.3	22
Northern	75.9	71.5	69.4
N Western	64.5	75.5	64.8
Southern	69.4	76.1	58.6
Western	75.8	33.5	73.6

Table 1:

Note 1.3.

1.: In this Data set we have four variables, province, incidence of poverty 1991, 1993, 1996
2.: Before data the was collected the incidence of poverty where unknown and the variables would be random variables. But after the data were collected we have observed values of random variables.
3.: The columns represent vectors, the rows also vectors.

\(\bullet \) A matrix written in terms of its sub-matrices is called a partitioned matrix.

Let \(A_{11}\), \(A_{12}\) , \(A_{21}\) and \(A_{22}\) be sub-matrices of matrix \(A\) such that \(A_{r\times s}\) has elements \(a_{ij},\\\qquad i=1,2,\ldots r\qquad j=1,2,\ldots ,s\) then we write \[A_{n\times p} = \begin {pmatrix} A_{r\times s} & A_{r\times (p-s)}\\ &\\ A_{(n-r)\times s} & A_{(n-r)\times (p-s)}\\ \end {pmatrix} \]

Example 1.4.

1.: If \(B = \begin {pmatrix} 13 & -4 & 2\\ 3 & 12 & 0\\ 2 & -2 & 10\\ \end {pmatrix} \) then one possible partition of \(B\) could be
\[B_{11} = \begin {pmatrix} 13 & -4\\ 3 & 12\\ \end {pmatrix}_{2\times 2}, B_{12} = \begin {pmatrix} 2\\ 0\\ \end {pmatrix}_{2\times 1}, B_{21} = \begin {bmatrix} 2, & -2\\ \end {bmatrix}_{1\times 2}, B_{22} = \begin {bmatrix} 10\\ \end {bmatrix}_{1\times 1} \]
2.: In the poverty data, let \(A_{11}\) consists of 2 rows consisting of Lusaka and Copperbelt one column for 1991 data. i.e \(B_{11}\) is \(2\times 1\) matrix determine \(B_{12}\) , \(B_{21}\) and \(B_{22}\).

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