LeviCivita | The totally anti-symmetric Levi Civita tensor |
Permutations | Form all permutations of a list |
InProduct | Inner product of vectors |
CrossProduct | Outer product of vectors |
ZeroVector | Create a vector with all zeroes |
BaseVector | Base vector |
Identity | Identity matrix |
ZeroMatrix | Matrix filled with zeroes |
DiagonalMatrix | Construct a diagonal matrix |
IsMatrix | Test whether argument is a matrix |
Normalize | Normalize a vector |
Transpose | Transpose of a matrix |
Determinant | Determinant of a matrix |
Trace | Trace of a matrix |
Inverse | Inverse of a matrix |
Minor | Principal minor of a matrix |
CoFactor | Cofactor of a matrix |
SolveMatrix | Solve a linear system |
CharacteristicEquation | Characteristic polynomial of a matrix |
EigenValues | Eigenvalues of a matrix |
EigenVectors | Eigenvectors of a matrix |
IsHermitean | Test whether a matrix is Hermitean |
IsUnitary | Test whether a matrix is unitary |
In> LeviCivita({1,2,3}) Out> 1; In> LeviCivita({2,1,3}) Out> -1; In> LeviCivita({2,2,3}) Out> 0; |
In> Permutations({a,b,c}) Out> {{a,b,c},{a,c,b},{c,a,b},{b,a,c},{b,c,a},{c,b,a}}; |
In> {a,b,c} . {d,e,f}; Out> a*d+b*e+c*f; |
In> {a,b,c} X {d,e,f}; Out> {b*f-c*e,c*d-a*f,a*e-b*d}; |
In> ZeroVector(4) Out> {0,0,0,0}; |
In> BaseVector(2,4) Out> {0,1,0,0}; |
In> Identity(3) Out> {{1,0,0},{0,1,0},{0,0,1}}; |
In> ZeroMatrix(3,4) Out> {{0,0,0,0},{0,0,0,0},{0,0,0,0}}; |
In> DiagonalMatrix(1 .. 4) Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}}; |
In> IsMatrix(ZeroMatrix(3,4)) Out> True; In> IsMatrix(ZeroVector(4)) Out> False; In> IsMatrix(3) Out> False; |
In> Normalize({3,4}) Out> {3/5,4/5}; In> % . % Out> 1; |
In> Transpose({{a,b}}) Out> {{a},{b}}; |
In> DiagonalMatrix(1 .. 4) Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}}; In> Determinant(%) Out> 24; |
In> DiagonalMatrix(1 .. 4) Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}}; In> Trace(%) Out> 10; |
In> DiagonalMatrix({a,b,c}) Out> {{a,0,0},{0,b,0},{0,0,c}}; In> Inverse(%) Out> {{(b*c)/(a*b*c),0,0},{0,(a*c)/(a*b*c),0},{0,0,(a*b)/(a*b*c)}}; In> Simplify(%) Out> {{1/a,0,0},{0,1/b,0},{0,0,1/c}}; |
In> A := {{1,2,3}, {4,5,6}, {7,8,9}}; Out> {{1,2,3},{4,5,6},{7,8,9}}; In> PrettyForm(A); / \ | ( 1 ) ( 2 ) ( 3 ) | | | | ( 4 ) ( 5 ) ( 6 ) | | | | ( 7 ) ( 8 ) ( 9 ) | \ / Out> True; In> Minor(A,1,2); Out> -6; In> Determinant({{2,3}, {8,9}}); Out> -6; |
In> A := {{1,2,3}, {4,5,6}, {7,8,9}}; Out> {{1,2,3},{4,5,6},{7,8,9}}; In> PrettyForm(A); / \ | ( 1 ) ( 2 ) ( 3 ) | | | | ( 4 ) ( 5 ) ( 6 ) | | | | ( 7 ) ( 8 ) ( 9 ) | \ / Out> True; In> CoFactor(A,1,2); Out> 6; In> Minor(A,1,2); Out> -6; In> Minor(A,1,2) * (-1)^(1+2); Out> 6; |
In> A := {{1,2}, {3,4}}; Out> {{1,2},{3,4}}; In> v := {5,6}; Out> {5,6}; In> x := SolveMatrix(A, v); Out> {-4,9/2}; In> A * x; Out> {5,6}; |
In> DiagonalMatrix({a,b,c}) Out> {{a,0,0},{0,b,0},{0,0,c}}; In> CharacteristicEquation(%,x) Out> (a-x)*(b-x)*(c-x); In> Expand(%,x) Out> (b+a+c)*x^2-x^3-((b+a)*c+a*b)*x+a*b*c; |
In> M:={{1,2},{2,1}} Out> {{1,2},{2,1}}; In> EigenValues(M) Out> {3,-1}; |
In> M:={{1,2},{2,1}} Out> {{1,2},{2,1}}; In> e:=EigenValues(M) Out> {3,-1}; In> EigenVectors(M,e) Out> {{-ki2/ -1,ki2},{-ki2,ki2}}; |
In> IsHermitean({{0,I},{-I,0}}) Out> True; In> IsHermitean({{0,I},{2,0}}) Out> False; |
In> IsUnitary({{0,I},{-I,0}}) Out> True; In> IsUnitary({{0,I},{2,0}}) Out> False; |