Output from imlrank.sas
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Output from imlrank.sas |
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NOTE: Capture of log output started.
NOTE: %INCLUDE (level 1) file n:\psy6140\examples\iml\imlrank.sas is file
n:\psy6140\examples\iml\imlrank.sas.
898 +title 'IMLRANK: Matrix rank and linear independence'; 899 +proc iml;
IML Ready
900 + *-- Define a function module to find the rank of a matrix; 901 +start r(A); 902 + reset noprint; 903 + *-- rank of a matrix (cant call it rank, since that name 904 + is used for sorting); 905 + *-- rank = number of nonzero rows/cols in echelon form; 906 + e = echelon(A); 907 + do i=1 to nrow(e); 907 + *-- Find rows which are not all zero; 908 + if any( e[i,] <> 0 ) then rows = rows || i; 909 + end; 910 + e = e[rows,]; 910 + *-- Keep only non-zero rows; 911 + do i=1 to ncol(e); 911 + *-- Find cols which are not all zero; 912 + if any( e[,i] <> 0 ) then cols = cols || i; 913 + end; 914 + e = e[,cols]; 914 + *-- Keep only non-zero cols; 915 + reset print; 916 + return( min( nrow(e), ncol(e)) ); 917 + finish;
NOTE: Module R defined.
918 +
919 + reset print log fuzz;
920 +* 1. MATRIX RANK AND LINEAR INDEPENDENCE;
921 +* A set of vectors is linearly DEPENDENT if there are scalars;
922 +* (not all =0) which give a linear combination = zero vector.;
923 +X1 = {1 3 5};
X1 1 row 3 cols (numeric)
1 3 5
924 +X2 = {0 1 2};
X2 1 row 3 cols (numeric)
0 1 2
925 +X3 = {-1 4 9};
X3 1 row 3 cols (numeric)
-1 4 9
926 +X4 = {2 2 2};
X4 1 row 3 cols (numeric)
2 2 2
927 +comb = (2#X1) + (-4#X2) + (0#X3) + -1#X4;
COMB 1 row 3 cols (numeric)
0 0 0
928 +* X3, X4 are linear combinations of X1, X2; 929 +print (t(X3)) '=' (t(-X1)) '+' (t(7#X2));
#TEM1001 #TEM1003 #TEM1005
-1 = -1 + 0
4 -3 7
9 -5 14
930 +print (t(X4)) '=' (t(2#X1)) '+' (t(-4#X2));
#TEM1001 #TEM1003 #TEM1006
2 = 2 + 0
2 6 -4
2 10 -8
931 +* only two of x1, x2, x3, x4 are linearly independent; 932 +X = t(X1) || t(X2) || t(X3) || t(X4);
X 3 rows 4 cols (numeric)
1 0 -1 2
3 1 4 2
5 2 9 2
933 +r = r(X);
R 1 row 1 col (numeric)
2
934 +* rank = number of nonzero rows in echelon form; 935 +r = echelon(X);
R 3 rows 4 cols (numeric)
1 0 -1 2
0 1 7 -4
0 0 0 0
936 +* change last element so X4 not dependent; 937 +X[3,4] = 4;
X 3 rows 4 cols (numeric)
1 0 -1 2
3 1 4 2
5 2 9 4
938 +r = r(X);
R 1 row 1 col (numeric)
3
939 +r = echelon(X);
R 3 rows 4 cols (numeric)
1 0 -1 0
0 1 7 0
0 0 0 1
941 +* -----------------------------------------------------;
942 +* 2. MATRIX RANK BY ELEMENTARY ROW OPERATIONS;
943 +* Use elementary row ops to reduce matrix to echelon
944 +* form. Zero rows/cols do not contribute to rank.;
945 +* -----------------------------------------------------;
946 +A = {4 1 8, 5 2 7, 5 1 11, 8 3 12};
A 4 rows 3 cols (numeric)
4 1 8
5 2 7
5 1 11
8 3 12
947 +SAVE = A;
SAVE 4 rows 3 cols (numeric)
4 1 8
5 2 7
5 1 11
8 3 12
948 +* ROW 4 - 2#ROW 1; 949 +A[4,] = A[4,] - 2#A[1,];
A 4 rows 3 cols (numeric)
4 1 8
5 2 7
5 1 11
0 1 -4
950 +* Each elementary row operation is equivalent to premultiplication 951 + by an identity matrix with the same operation applied to its rows; 952 +T = I(4);
T 4 rows 4 cols (numeric)
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
953 +T[4,] = T[4,] - 2#T[1,];
T 4 rows 4 cols (numeric)
1 0 0 0
0 1 0 0
0 0 1 0
-2 0 0 1
954 +A = T * SAVE;
A 4 rows 3 cols (numeric)
4 1 8
5 2 7
5 1 11
0 1 -4
955 + * ROW 3 - ROW 2; 956 +A[3,] = A[3,] - A[2,];
A 4 rows 3 cols (numeric)
4 1 8
5 2 7
0 -1 4
0 1 -4
957 + * .25 # ROW 1; 958 +A[1,] = .25 # A[1,];
A 4 rows 3 cols (numeric)
1 0.25 2
5 2 7
0 -1 4
0 1 -4
959 + * ROW 2 - 5#ROW 1; 960 +A[2,] = A[2,] - 5#A[1,];
A 4 rows 3 cols (numeric)
1 0.25 2
0 0.75 -3
0 -1 4
0 1 -4
961 + * ROW 4 + 1#ROW 3; 962 +A[4,] = A[4,] + A[3,];
A 4 rows 3 cols (numeric)
1 0.25 2
0 0.75 -3
0 -1 4
0 0 0
963 + * 1.33#ROW 2; 964 +A[2,] = (4/3) # A[2,];
A 4 rows 3 cols (numeric)
1 0.25 2
0 1 -4
0 -1 4
0 0 0
965 + * ROW 2 + ROW 3; 966 +A[2,] = A[2,] + A[3,];
A 4 rows 3 cols (numeric)
1 0.25 2
0 0 0
0 -1 4
0 0 0
967 + *-- RANK = number of nonzero rows/cols; 968 +r = r(A);
R 1 row 1 col (numeric)
2
969 +quit;
Exiting IML.
NOTE: The PROCEDURE IML used 0.22 seconds.
970 +
NOTE: %INCLUDE (level 1) ending.