@deffn {Matrix Function} DET (@var{M})
Returns the determinant of square matrix @var{M}.
+
+@t{DET(@{3, 7; 1, -4@}) @result{} -19}
@end deffn
@deffn {Matrix Function} EVAL (@var{M})
Use @code{CALL EIGEN} (@pxref{CALL EIGEN}) to compute eigenvalues and
eigenvectors of a matrix.
+
+@t{EVAL(@{2, 0, 0; 0, 3, 4; 0, 4, 9@}) @result{} @{11; 2; 1@}}
@end deffn
@deffn {Matrix Function} GINV (@var{M})
@var{M}, defined such that
@math{@var{M}@times{}@var{A}@times{}@var{M}=@var{M}} and
@math{@var{A}@times{}@var{M}@times{}@var{A}=@var{A}}.
-@end deffn
+@t{GINV(@{1, 2@}) @result{} @{.2; .4@}} (approximately) @*
+@t{@{1:9@} * GINV(1:9) * @{1:9@} @result{} @{1:9@}} (approximately)
+@end deffn
@deffn {Matrix Function} GSCH (@var{M})
@var{M} must be a @math{@var{n}@times{}@var{m}} matrix, @math{@var{m}
@geq{} @var{n}}, with rank @var{n}. Returns an
@math{@var{n}@times{}@var{n}} orthonormal basis for @var{M}, obtained
using the Gram-Schmidt process.
+
+@t{GSCH(@{3, 2; 1, 2@}) * SQRT(10) @result{} @{3, -1; 1, 3@}} (approximately)
@end deffn
@deffn {Matrix Function} INV (@var{M})
that @math{@var{M}@times{}@var{A} = @var{A}@times{}@var{M} = I}, where
@var{I} is the identity matrix. @var{M} must not be singular, that
is, @math{\det(@var{M}) @ne{} 0}.
+
+@t{INV(@{4, 7; 2, 6@}) @result{} @{.6, -.7; -.2, .4@}} (approximately)
@end deffn
@deffn {Matrix Function} KRONEKER (@var{Ma}, @var{Mb})
product of @var{Mb} by a different element of @var{Ma}. For example,
when @code{A} is a @math{2@times{}2} matrix, @code{KRONEKER(A, B)} is
equivalent to @code{@{A(1,1)*B, A(1,2)*B; A(2,1)*B, A(2,2)*B@}}.
+
+@format
+@t{KRONEKER(@{1, 2; 3, 4@}, @{0, 5; 6, 7@}) @result{}
+ 0 5 0 10
+ 6 7 12 14
+ 0 15 0 20
+ 18 21 24 28}
+@end format
@end deffn
@deffn {Matrix Function} RANK (@var{M})
Returns the rank of matrix @var{M}, a integer scalar whose value is
the dimension of the vector space spanned by its columns or,
equivalently, by its rows.
+
+@format
+@t{RANK(@{1, 0, 1; -2, -3, 1; 3, 3, 0@}) @result{} 2
+RANK(@{1, 1, 0, 2; -1, -1, 0, -2@}) @result{} 1
+RANK(@{1, -1; 1, -1; 0, 0; 2, -2@}) @result{} 1
+RANK(@{1, 2, 1; -2, -3, 1; 3, 5, 0@}) @result{} 2
+RANK(@{1, 0, 2; 2, 1, 0; 3, 2, 1@}) @result{} 3}
+@end format
@end deffn
@deffn {Matrix Function} SOLVE (@var{Ma}, @var{Mb})
@math{@var{n}@times{}@var{k}} matrix. Returns an
@math{@var{n}@times{}@var{k}} matrix @var{X} such that @math{@var{Ma}
@times{} @var{X} = @var{Mb}}.
+
+All of the following examples show approximate results:
+
+@format
+@t{SOLVE(@{2, 3; 4, 9@}, @{6, 2; 15, 5@}) @result{}
+ 1.50 .50
+ 1.00 .33
+SOLVE(@{1, 3, -2; 3, 5, 6; 2, 4, 3@}, @{5; 7; 8@}) @result{}
+ -15.00
+ 8.00
+ 2.00
+SOLVE(@{2, 1, -1; -3, -1, 2; -2, 1, 2@}, @{8; -11; -3@}) @result{}
+ 2.00
+ 3.00
+ -1.00}
+@end format
@end deffn
@deffn {Matrix Function} SVAL (@var{M})
Use @code{CALL SVD} (@pxref{CALL SVD}) to compute the full singular
value decomposition of a matrix.
+
+@format
+@t{SVAL(@{1, 1; 0, 0@}) @result{} @{1.41; .00@}
+SVAL(@{1, 0, 1; 0, 1, 1; 0, 0, 0@}) @result{} @{1.73; 1.00; .00@}
+SVAL(@{2, 4; 1, 3; 0, 0; 0, 0@}) @result{} @{5.46; .37@}}
+@end format
@end deffn
@deffn {Matrix Function} SWEEP (@var{M}, @var{nk})
@math{@var{A}_{kk} = 1/@var{M}_{kk}},
@math{@var{A}_{ik} = -@var{M}_{ik}/@var{M}_{kk} @r{for} i @ne{} k},
@math{@var{A}_{kj} = @var{M}_{kj}/@var{M}_{kk} @r{for} j @ne{} k, @r{and}}
-@math{@var{A}_{ij} = @var{M}_{ij} - (@var{M}_{ik} * @var{M}_{kj}) / @var{M}_{kk} @r{for} i @ne{} k @r{and} j @ne{} k}.
+@math{@var{A}_{ij} = @var{M}_{ij} - @var{M}_{ik}@var{M}_{kj}/@var{M}_{kk} @r{for} i @ne{} k @r{and} j @ne{} k}.
@end display
If @math{@var{M}_{kk} = 0}, then:
@math{@var{A}_{ik} = @var{A}_{ki} = 0 @r{and}}
@math{@var{A}_{ij} = @var{M}_{ij}, @r{for} i @ne{} k @r{and} j @ne{} k}.
@end display
+
+Given @t{M = @{0, 1, 2; 3, 4, 5; 6, 7, 8@}}, then (approximately):
+
+@format
+@t{SWEEP(M, 1) @result{}
+ .00 .00 .00
+ .00 4.00 5.00
+ .00 7.00 8.00
+SWEEP(M, 2) @result{}
+ -.75 -.25 .75
+ .75 .25 1.25
+ .75 -1.75 -.75
+SWEEP(M, 3) @result{}
+ -1.50 -.75 -.25
+ -.75 -.38 -.63
+ .75 .88 .13}
+@end format
@end deffn
@node Matrix IO
-@subsubsection I/O
+@subsubsection I/O Function
+
+This function works with files being used on the @code{READ} statement.
@deffn {Matrix Function} EOF (@var{file})
@anchor{EOF Matrix Function}
Given a file handle or file name @var{file}, returns an integer scalar
-that indicates whether the last record in the file has been read.
-Determining this requires attempting reading past the current record,
+1 if the last line in the file has been read or 0 if more lines are
+available. Determining this requires attempting to read another line,
which means that @code{REREAD} on the next @code{READ} command
-following @code{EOF} on the same file will be ineffective.
+following @code{EOF} on the same file will be ineffective.
@end deffn
+The @code{EOF} function gives a matrix program the flexibility to read
+a file with text data without knowing the length of the file in
+advance. For example, the following program will read all the lines
+of data in @file{data.txt}, each consisting of three numbers, as rows
+in matrix @code{data}:
+
+@verbatim
+MATRIX.
+COMPUTE data={}.
+LOOP IF NOT EOF('data.txt').
+ READ row/FILE='data.txt'/FIELD=1 TO 1000/SIZE={1,3}.
+ COMPUTE data={data; row}.
+END LOOP.
+PRINT data.
+END MATRIX.
+
+@end verbatim
+
@node Matrix COMPUTE Command
@subsection The @code{COMPUTE} Command
Use the @code{EVAL} function (@pxref{EVAL}) to compute just the
eigenvalues of a symmetric matrix.
+For example, the following matrix language commands:
+@example
+CALL EIGEN(@{1, 0; 0, 1@}, evec, eval).
+PRINT evec.
+PRINT eval.
+
+CALL EIGEN(@{3, 2, 4; 2, 0, 2; 4, 2, 3@}, evec2, eval2).
+PRINT evec2.
+PRINT eval2.
+@end example
+
+@noindent
+yield this output:
+
+@example
+evec
+ 1 0
+ 0 1
+
+eval
+ 1
+ 1
+
+evec2
+ -.6666666667 .0000000000 .7453559925
+ -.3333333333 -.8944271910 -.2981423970
+ -.6666666667 .4472135955 -.5962847940
+
+eval2
+ 8.0000000000
+ -1.0000000000
+ -1.0000000000
+@end example
+
@item @t{CALL SVD(@var{M}, @var{U}, @var{S}, @var{V})}
@anchor{CALL SVD}
Use the @code{SVAL} function (@pxref{SVAL}) to compute just the
singular values of a matrix.
+
+For example, the following matrix program:
+
+@example
+CALL SVD(@{3, 2, 2; 2, 3, -2@}, u, s, v).
+PRINT (u * s * T(v))/FORMAT F5.1.
+@end example
+
+@noindent
+yields this output:
+
+@example
+(u * s * T(v))
+ 3.0 2.0 2.0
+ 2.0 3.0 -2.0
+@end example
@end table
The final procedure is implemented via @code{CALL} to allow it to
Use the @code{MDIAG} function (@pxref{MDIAG}) to construct a new
matrix with a specified main diagonal.
+
+For example, this matrix program:
+
+@example
+COMPUTE x=@{1, 2, 3; 4, 5, 6; 7, 8, 9@}.
+CALL SETDIAG(x, 10).
+PRINT x.
+@end example
+
+@noindent
+outputs the following:
+
+@example
+x
+ 10 2 3
+ 4 10 6
+ 7 8 10
+@end example
@end table
@node Matrix PRINT Command
@enumerate
@item
-If the matrix's elements are all integers, then, if possible, @pspp{}
-chooses the narrowest @code{F} format with width 12 or less that fits
-@var{m} plus a sign.
+If @math{@var{m} < 10^{11}} and the matrix's elements are all
+integers, @pspp{} chooses the narrowest @code{F} format that fits
+@var{m} plus a sign. For example, if the matrix is @t{@{1:10@}}, then
+@math{m = 10}, which fits in 3 columns with room for a sign, the
+format is @code{F3.0}.
@item
Otherwise, if @math{@var{m} @geq{} 10^9} or @math{@var{m} @leq{}
10^{-4}}, @pspp{} scales all of the numbers in the matrix by
-@math{10^x}, where @var{x} is the exponent used when @var{m} is
-displayed in scientific notation, and displays the scaled value in
-format @code{F13.10}. @pspp{} adds a note to the output to indicate
-the scale factor.
+@math{10^x}, where @var{x} is the exponent that would be used to
+display @var{m} in scientific notation. For example, for
+@math{@var{m} = 5.123@times{}10^{20}}, the scale factor is
+@math{10^{20}}. @pspp{} displays the scaled values in format
+@code{F13.10} and notes the scale factor in the output.
@item
-Otherwise, @pspp{} displays the value, without scaling, in format
-@code{F13.10}.
+Otherwise, @pspp{} displays the matrix values, without scaling, in
+format @code{F13.10}.
@end enumerate
The optional @code{TITLE} subcommand specifies a title for the output
The @code{CLABELS} and @code{CNAMES} subcommands work for labeling
columns as @code{RLABELS} and @code{RNAMES} do for labeling rows.
-@subsubheading Text Output
-
When the @var{expression} is omitted, @code{PRINT} does not output a
matrix. Instead, it outputs only the text specified on @code{TITLE},
if any, preceded by any space specified on the @code{SPACE}
command acts as if @code{MDISPLAY} is set to @code{TEXT} regardless of
its actual setting.
+
+
@node Matrix DO IF Command
@subsection The @code{DO IF} Command
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