Dot, . | get dot product of tensors |
InProduct | inner product of vectors (deprecated) |
CrossProduct | outer product of vectors |
Outer, o | get outer tensor product |
ZeroVector | create a vector with all zeroes |
BaseVector | base vector |
Identity | make identity matrix |
ZeroMatrix | make a zero matrix |
Diagonal | extract the diagonal from a matrix |
DiagonalMatrix | construct a diagonal matrix |
OrthogonalBasis | create an orthogonal basis |
OrthonormalBasis | create an orthonormal basis |
Normalize | normalize a vector |
Transpose | get transpose of a matrix |
Determinant | determinant of a matrix |
Trace | trace of a matrix |
Inverse | get inverse of a matrix |
Minor | get principal minor of a matrix |
CoFactor | cofactor of a matrix |
MatrixPower | get nth power of a square matrix |
SolveMatrix | solve a linear system |
CharacteristicEquation | get characteristic polynomial of a matrix |
EigenValues | get eigenvalues of a matrix |
EigenVectors | get eigenvectors of a matrix |
Sparsity | get the sparsity of a matrix |
Cholesky | find the Cholesky Decomposition |
Dot(t1,t2) t1 . t2 |
In> Dot({1,2},{3,4}) Out> 11; In> Dot({{1,2},{3,4}},{5,6}) Out> {17,39}; In> Dot({5,6},{{1,2},{3,4}}) Out> {23,34}; In> Dot({{1,2},{3,4}},{{5,6},{7,8}}) Out> {{19,22},{43,50}}; |
Or, using the "."-Operator: |
In> {1,2} . {3,4} Out> 11; In> {{1,2},{3,4}} . {5,6} Out> {17,39}; In> {5,6} . {{1,2},{3,4}} Out> {23,34}; In> {{1,2},{3,4}} . {{5,6},{7,8}} Out> {{19,22},{43,50}}; |
InProduct(a,b) |
This function is superceded by the . operator.
In> {a,b,c} . {d,e,f}; Out> a*d+b*e+c*f; |
CrossProduct(a,b) a X b |
In> {a,b,c} X {d,e,f}; Out> {b*f-c*e,c*d-a*f,a*e-b*d}; |
Outer(t1,t2) t1 o t2 |
In> Outer({1,2},{3,4,5}) Out> {{3,4,5},{6,8,10}}; In> Outer({a,b},{c,d}) Out> {{a*c,a*d},{b*c,b*d}}; |
Or, using the "o"-Operator: |
In> {1,2} o {3,4,5} Out> {{3,4,5},{6,8,10}}; In> {a,b} o {c,d} Out> {{a*c,a*d},{b*c,b*d}}; |
ZeroVector(n) |
In> ZeroVector(4) Out> {0,0,0,0}; |
BaseVector(k, n) |
n -- dimension of the vector
In> BaseVector(2,4) Out> {0,1,0,0}; |
Identity(n) |
In> Identity(3) Out> {{1,0,0},{0,1,0},{0,0,1}}; |
ZeroMatrix(n) ZeroMatrix(n, m) |
m -- number of columns
In> ZeroMatrix(3,4) Out> {{0,0,0,0},{0,0,0,0},{0,0,0,0}}; In> ZeroMatrix(3) Out> {{0,0,0},{0,0,0},{0,0,0}}; |
Diagonal(A) |
In> Diagonal(5*Identity(4)) Out> {5,5,5,5}; In> Diagonal(HilbertMatrix(3)) Out> {1,1/3,1/5}; |
DiagonalMatrix(d) |
In> DiagonalMatrix(1 .. 4) Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}}; |
OrthogonalBasis(W) |
In> OrthogonalBasis({{1,1,0},{2,0,1},{2,2,1}}) Out> {{1,1,0},{1,-1,1},{-1/3,1/3,2/3}}; |
OrthonormalBasis(W) |
In> OrthonormalBasis({{1,1,0},{2,0,1},{2,2,1}}) Out> {{Sqrt(1/2),Sqrt(1/2),0},{Sqrt(1/3),-Sqrt(1/3),Sqrt(1/3)}, {-Sqrt(1/6),Sqrt(1/6),Sqrt(2/3)}}; |
Normalize(v) |
In> v:=Normalize({3,4}) Out> {3/5,4/5}; In> v . v Out> 1; |
Transpose(M) |
In> Transpose({{a,b}}) Out> {{a},{b}}; |
Determinant(M) |
In> A:=DiagonalMatrix(1 .. 4) Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}}; In> Determinant(A) Out> 24; |
Trace(M) |
In> A:=DiagonalMatrix(1 .. 4) Out> {{1,0,0,0},{0,2,0,0},{0,0,3,0},{0,0,0,4}}; In> Trace(A) Out> 10; |
Inverse(M) |
In> A:=DiagonalMatrix({a,b,c}) Out> {{a,0,0},{0,b,0},{0,0,c}}; In> B:=Inverse(A) Out> {{(b*c)/(a*b*c),0,0},{0,(a*c)/(a*b*c),0}, {0,0,(a*b)/(a*b*c)}}; In> Simplify(B) Out> {{1/a,0,0},{0,1/b,0},{0,0,1/c}}; |
Minor(M,i,j) |
i, j - positive integers
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; |
CoFactor(M,i,j) |
i, j - positive integers
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; |
MatrixPower(mat,n) |
n -- an integer
In> A:={{1,2},{3,4}} Out> {{1,2},{3,4}}; In> MatrixPower(A,0) Out> {{1,0},{0,1}}; In> MatrixPower(A,1) Out> {{1,2},{3,4}}; In> MatrixPower(A,3) Out> {{37,54},{81,118}}; In> MatrixPower(A,-3) Out> {{-59/4,27/4},{81/8,-37/8}}; |
SolveMatrix(M,v) |
v -- a vector
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}; |
CharacteristicEquation(matrix,var) |
var -- a free variable
In> A:=DiagonalMatrix({a,b,c}) Out> {{a,0,0},{0,b,0},{0,0,c}}; In> B:=CharacteristicEquation(A,x) Out> (a-x)*(b-x)*(c-x); In> Expand(B,x) Out> (b+a+c)*x^2-x^3-((b+a)*c+a*b)*x+a*b*c; |
EigenValues(matrix) |
It first determines the characteristic equation, and then factorizes this equation, returning the roots of the characteristic equation Det(matrix-x*identity).
In> M:={{1,2},{2,1}} Out> {{1,2},{2,1}}; In> EigenValues(M) Out> {3,-1}; |
EigenVectors(A,eigenvalues) |
eigenvalues -- list of eigenvalues as returned by EigenValues
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}}; |
Sparsity(matrix) |
In> Sparsity(Identity(2)) Out> 0.5; In> Sparsity(Identity(10)) Out> 0.9; In> Sparsity(HankelMatrix(10)) Out> 0.45; In> Sparsity(HankelMatrix(100)) Out> 0.495; In> Sparsity(HilbertMatrix(10)) Out> 0; In> Sparsity(ZeroMatrix(10,10)) Out> 1; |
Cholesky(A) |
In> A:={{4,-2,4,2},{-2,10,-2,-7},{4,-2,8,4},{2,-7,4,7}} Out> {{4,-2,4,2},{-2,10,-2,-7},{4,-2,8,4},{2,-7,4,7}}; In> R:=Cholesky(A); Out> {{2,-1,2,1},{0,3,0,-2},{0,0,2,1},{0,0,0,1}}; In> Transpose(R)*R = A Out> True; In> Cholesky(4*Identity(5)) Out> {{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0},{0,0,0,0,2}}; In> Cholesky(HilbertMatrix(3)) Out> {{1,1/2,1/3},{0,Sqrt(1/12),Sqrt(1/12)},{0,0,Sqrt(1/180)}}; In> Cholesky(ToeplitzMatrix({1,2,3})) In function "Check" : CommandLine(1) : "Cholesky: Matrix is not positive definite" |