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Unformatted text preview: PubH 7405: EGRESSION ANALYSIS A REVIEW OF MATRICES Scientists use statistics in their research – for the design and analysis of their studies. They consult statisticians when needed; and, in doing so, they need to understand some statistical reasoning that underlies the method they employ . Statisticians are concerned with the applications of existing statistical methods – and also with devising new methods from time to time; and, in doing so, they need to know enough mathematics for the tasks . One of such areas of mathematics is “ matrices ”. The roles of mathematics in statistics is similar to that of statistics in science as a whole. Statistics is the “language” for scientists and Mathematics is the “language” for statisticians. A “ matrix ” is a display of numbers or numerical quantities laid out in a rectangular array of rows and columns . The array, or twoway table of numbers , could be rectangular or square – could be just one row (a row matrix or row vector ) or one column (a column matrix or column vector ). When it is square, it could be symmetric or a “ diagonal matrix ” (nonzero entries are on the main diagonal). The numbers of rows and of columns form the “ dimension ” of a matrix; for example, the dimension of the following matrix is “3x2” – three rows and two columns . 35 21,000 47 33,000 23 16,000 A number is the most simple matrix; a 1x1 matrix, one row one column – called a “ scalar ” An “entry” or “element” of a matrix need two subscripts for identification; the first for the row number and the second for the column number: ] [ ij a A For example, in the following matrix we have a 11 = 16,000 and a 32 = 35 . 35 21,000 47 33,000 23 16,000 A Two matrices A and B are equal if and only if they have the same dimension and all corresponding elements are equal – position by position } j all and i all for , , { ] [ ] [ ij ij nxm ij rxc ij b a m c n r b a B A B A Again, a “ vector ” is a matrix with only one row (a row vector ) or with only one column (a column vector ). It is conventional to assume that all vectors are column vectors – unless otherwise stated. n c c c 2 1 C A “ triangular matrix ” is a square matrix with all the elements above, or below, the main diagonal equal to 0; called “lower triangular matrix” and “upper triangular matrix”. For example, 11 22 9 6 7 2 A A square matrix whose elements are all zero except for those on the main diagonal is called a diagonal matrix . If all diagonal elements of a diagonal matrix are equal , the matrix is call a “ scalar matrix ”: 9 9 9 D In particular, a scalar matrix with the common diagonal element equal to 1 is a “ unit matrix ” and is denoted by “...
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This note was uploaded on 11/21/2011 for the course PUBH 7405 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff

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