| IsScalar | test for a scalar |
| IsVector | test for a vector |
| IsMatrix | test for a matrix |
| IsSquareMatrix | test for a square matrix |
| IsHermitian | test for a Hermitian matrix |
| IsOrthogonal | test for an orthogonal matrix |
| IsDiagonal | test for a diagonal matrix |
| IsLowerTriangular | test for a lower triangular matrix |
| IsUpperTriangular | test for an upper triangular matrix |
| IsSymmetric | test for a symmetric matrix |
| IsSkewSymmetric | test for a skew-symmetric matrix |
| IsUnitary | test for a unitary matrix |
| IsIdempotent | test for an idempotent matrix |
IsScalar(expr) |
In> IsScalar(7)
Out> True;
In> IsScalar(Sin(x)+x)
Out> True;
In> IsScalar({x,y})
Out> False;
|
IsVector(expr) |
IsVector(pred,expr) |
pred -- predicate test (e.g. IsNumber, IsInteger, ...)
In> IsVector({a,b,c})
Out> True;
In> IsVector({a,{b},c})
Out> False;
In> IsVector(IsInteger,{1,2,3})
Out> True;
In> IsVector(IsInteger,{1,2.5,3})
Out> False;
|
IsMatrix(expr) |
IsMatrix(pred,expr) |
pred -- predicate test (e.g. IsNumber, IsInteger, ...)
In> IsMatrix(1)
Out> False;
In> IsMatrix({1,2})
Out> False;
In> IsMatrix({{1,2},{3,4}})
Out> True;
In> IsMatrix(IsRational,{{1,2},{3,4}})
Out> False;
In> IsMatrix(IsRational,{{1/2,2/3},{3/4,4/5}})
Out> True;
|
IsSquareMatrix(expr) |
IsSquareMatrix(pred,expr) |
pred -- predicate test (e.g. IsNumber, IsInteger, ...)
In> IsSquareMatrix({{1,2},{3,4}});
Out> True;
In> IsSquareMatrix({{1,2,3},{4,5,6}});
Out> False;
In> IsSquareMatrix(IsBoolean,{{1,2},{3,4}});
Out> False;
In> IsSquareMatrix(IsBoolean,{{True,False},{False,True}});
Out> True;
|
IsHermitian(A) |
In> IsHermitian({{0,I},{-I,0}})
Out> True;
In> IsHermitian({{0,I},{2,0}})
Out> False;
|
IsOrthogonal(A) |
In> A := {{1,2,2},{2,1,-2},{-2,2,-1}};
Out> {{1,2,2},{2,1,-2},{-2,2,-1}};
In> PrettyForm(A/3)
|
/ \ | / 1 \ / 2 \ / 2 \ | | | - | | - | | - | | | \ 3 / \ 3 / \ 3 / | | | | / 2 \ / 1 \ / -2 \ | | | - | | - | | -- | | | \ 3 / \ 3 / \ 3 / | | | | / -2 \ / 2 \ / -1 \ | | | -- | | - | | -- | | | \ 3 / \ 3 / \ 3 / | \ / Out> True; In> IsOrthogonal(A/3) Out> True; |
IsDiagonal(A) |
In> IsDiagonal(Identity(5)) Out> True; In> IsDiagonal(HilbertMatrix(5)) Out> False; |
IsLowerTriangular(A) IsUpperTriangular(A) |
IsLowerTriangular(A) returns True if A is a lower triangular matrix and False otherwise. IsUpperTriangular(A) returns True if A is an upper triangular matrix and False otherwise.
In> IsUpperTriangular(Identity(5))
Out> True;
In> IsLowerTriangular(Identity(5))
Out> True;
In> IsLowerTriangular({{1,2},{0,1}})
Out> False;
In> IsUpperTriangular({{1,2},{0,1}})
Out> True;
|
In> IsUpperTriangular({{1,2,3},{0,1,2}})
Out> False;
|
IsSymmetric(A) |
In> A := {{1,0,0,0,1},{0,2,0,0,0},{0,0,3,0,0},
{0,0,0,4,0},{1,0,0,0,5}};
In> PrettyForm(A)
|
/ \ | ( 1 ) ( 0 ) ( 0 ) ( 0 ) ( 1 ) | | | | ( 0 ) ( 2 ) ( 0 ) ( 0 ) ( 0 ) | | | | ( 0 ) ( 0 ) ( 3 ) ( 0 ) ( 0 ) | | | | ( 0 ) ( 0 ) ( 0 ) ( 4 ) ( 0 ) | | | | ( 1 ) ( 0 ) ( 0 ) ( 0 ) ( 5 ) | \ / Out> True; In> IsSymmetric(A) Out> True; |
IsSkewSymmetric(A) |
In> A := {{0,-1},{1,0}}
Out> {{0,-1},{1,0}};
In> PrettyForm(%)
|
/ \ | ( 0 ) ( -1 ) | | | | ( 1 ) ( 0 ) | \ / Out> True; In> IsSkewSymmetric(A); Out> True; |
IsUnitary(A) |
A matrix A is orthogonal iff A^(-1) = Transpose( Conjugate(A) ). This is equivalent to the fact that the columns of A build an orthonormal system (with respect to the scalar product defined by InProduct).
In> IsUnitary({{0,I},{-I,0}})
Out> True;
In> IsUnitary({{0,I},{2,0}})
Out> False;
|
IsIdempotent(A) |
In> IsIdempotent(ZeroMatrix(10,10)); Out> True; In> IsIdempotent(Identity(20)) Out> True; |