МАТРИЦИ И ОПЕРАЦИИ С МАТРИЦИ

{a,b,c}

m={{a,b},{c,d}}

m// MatrixForm

m[[1]]

m[[1,2]]

v={x,y}

p v+q

v+{xp,yp}+{xpp,ypp}

{x,y}.{xp,yp}

{x,y}{xp,yp}

m.v

v.m

m.m //MatrixForm

v.m.v

Dot[v,m,v]

Det[m]

Transpose[m]// MatrixForm

n=Inverse[m]

n . m //Simplify

m . n //Simplify

h3=Table[1/(i+j-1),{i,3},{j,3}]
Det[h3]

Inverse[h3]//MatrixForm

h4=Table[1/(i+j-1),{i,4},{j,4}]
Det[h4]

Inverse[h4]//MatrixForm

h3.Inverse[h3]
h4.Inverse[h4]

r=Table[i+j+1,{i,3},{j,3}]
Eigenvalues[r]
Eigenvectors[r]

rn=N[r]
Eigenvalues[rn]
Eigenvectors[rn]

mm=Array[a,{3,3}]

Minors[mm,2]

tr=Sum[m[[i,i]],{i,2}]

Tr[m]

MatrixPower[m,3]

m^3

mmm=(({{0.4, 0.6}, {0.525, 0.475}}))

MatrixPower[mmm,3]

mmm.mmm.mmm

MatrixExp[mmm]

Exp[mmm]

A=Table[i j Min[i,j],{i,6},{j,6}]

A //MatrixForm

(({{1, 2, 3, 4, 5, 6}, {2, 8, 12, 16, 20, 24}, {3, 12, 27, 36, 45, 54}, {4, 16, 36, 64, 80, 96}, {5, 20, 45, 80, 125, 150}, {6, 24, 54, 96, 150, 216}}))

(({{1, 2, 3, 4}, {2, 8, 12, 16}, {3, 12, 27, 36}, {4, 16, 36, 64}}))

Inverse[A] //MatrixForm

(({{2, -(1/2), 0, 0, 0, 0}, {-(1/2), (1/2), -(1/6), 0, 0, 0}, {0, -(1/6), (2/9), -(1/12), 0, 0}, {0, 0, -(1/12), (1/8), -(1/20), 0}, {0, 0, 0, -(1/20), (2/25), -(1/30)}, {0, 0, 0, 0, -(1/30), (1/36)}}))

(({{2, -(1/2), 0, 0}, {-(1/2), (1/2), -(1/6), 0}, {0, -(1/6), (2/9), -(1/12)}, {0, 0, -(1/12), (1/16)}}))

MatrixRank[A]

Det[A]

Eigenvalues[A]

{Root[518400-1531872 #1+993560 ((#1)^2)-219492 ((#1)^3)+17836 ((#1)^4)-441 ((#1)^5)+((#1)^6)&,6],Root[518400-1531872 #1+993560 ((#1)^2)-219492 ((#1)^3)+17836 ((#1)^4)-441 ((#1)^5)+((#1)^6)&,5],Root[518400-1531872 #1+993560 ((#1)^2)-219492 ((#1)^3)+17836 ((#1)^4)-441 ((#1)^5)+((#1)^6)&,4],Root[518400-1531872 #1+993560 ((#1)^2)-219492 ((#1)^3)+17836 ((#1)^4)-441 ((#1)^5)+((#1)^6)&,3],Root[518400-1531872 #1+993560 ((#1)^2)-219492 ((#1)^3)+17836 ((#1)^4)-441 ((#1)^5)+((#1)^6)&,2],Root[518400-1531872 #1+993560 ((#1)^2)-219492 ((#1)^3)+17836 ((#1)^4)-441 ((#1)^5)+((#1)^6)&,1]}

N[%]

{397.5032850980498`,25.926618845715986`,9.857141444986159`,5.090499088899524`,2.1579025994774916`,0.4645529228710743`}

f[i_,j_]:=Min[i,j];B=Array[f,{5,5}]

B //MatrixForm

(({{1, 1, 1, 1, 1}, {1, 2, 2, 2, 2}, {1, 2, 3, 3, 3}, {1, 2, 3, 4, 4}, {1, 2, 3, 4, 5}}))

MatrixForm[Inverse[B]]

(({{2, -1, 0, 0, 0}, {-1, 2, -1, 0, 0}, {0, -1, 2, -1, 0}, {0, 0, -1, 2, -1}, {0, 0, 0, -1, 1}}))

S=SparseArray[{{5,5}->-1}] // MatrixForm

(({{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, -1}}))

G=SparseArray[{Band[{1,1}]->2,Band[{2,1}]->-1,Band[{1,2}]->-1},{5,5}]//MatrixForm

(({{2, -1, 0, 0, 0}, {-1, 2, -1, 0, 0}, {0, -1, 2, -1, 0}, {0, 0, -1, 2, -1}, {0, 0, 0, -1, 2}}))

MatrixForm[IdentityMatrix[5]]

(({{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}}))

MatrixForm[DiagonalMatrix[{1,1/2,1/3,1/4,1/5}]]

(({{1, 0, 0, 0, 0}, {0, (1/2), 0, 0, 0}, {0, 0, (1/3), 0, 0}, {0, 0, 0, (1/4), 0}, {0, 0, 0, 0, (1/5)}}))

ChebyshevT[5,x]

5 x-20 (x^3)+16 (x^5)

Plot[5 x-20 (x^3)+16 (x^5),{x,-1,1}]

ChebyshevU[5,x]

6 x-32 (x^3)+32 (x^5)

LegendreP[6,x]

Plot[(1/16) (-5+105 (x^2)-315 (x^4)+231 (x^6)),{x,-1,1}]

Simplify[(1/16) (-5+105 (x^2)-315 (x^4)+231 (x^6))/.x->1]

1