Lösungen

1

m1 = {{1, 2}, {2, 1}};  Eigensystem[m1]

m2 = {{1, 0, 0}, {1, 2, 0}, {1, 2, 3}};  Eigensystem[m2]

m3 = {{1, -2, 2}, {-1, 1, 1}, {-1, -2, 4}};  
    Eigensystem[m3]

2

A = {{a, -1}, {1, -1}};  Eigensystem[A]

Plot[-3 + 2 a + (a^2), {a, -4, 3}];

Solve[-3 + 2 a + (a^2) == 0, {a}]

Ab = {{-3, -1}, {1, -1}};
v = α {-1, 
              1}  + β {1, -1};
Ab . v // Expand

Ac = {{4, -1}, {1, -1}};
v = α {-1, 
              1}  + β {1, -1};
Ac . v // Expand

3

v1 = {1, -1, 1}; v2 = {-1, -1, 2}; v3 = {1, 0, -1}; 
    h = {1, 10, 100};  B = Transpose[{v1, v2, v3}];  
    B // MatrixForm

a

Det[B]

b

B1=Transpose[{2 v1,-4 v2,v3}]; B1//MatrixForm

m4=B1.Inverse[B]; m4//MatrixForm

m4.h

c

B2=Transpose[{2 v1,-4 v2,v2}]; B2//MatrixForm

m5=B2.Inverse[B]; m5//MatrixForm

m5.h

4

u={2,-3,10}; a={1,-1,2}; b={1,0,-2}; n={0,0,0}; OQ={7,2,6};
matr1=Transpose[{u,a,b}];
matr2=Transpose[{n,a,b}];
Print[MatrixForm[matr1]];
Print[MatrixForm[matr2]];

a

proj= matr2.Inverse[matr1]; MatrixForm[proj]

b

OQ1=proj.OQ

c

OQ2={OQ1[[1]],OQ1[[2]]}

α= Pi/6;
d={{Cos[α],-Sin[α]},{Sin[α],Cos[α]}};
d// MatrixForm

%//N

OQ3=d.OQ2

%//N

5

X = 
        Inverse[Transpose[B]] . m3 . Transpose[B] . 
          Inverse[m3];
X // MatrixForm

% // N // MatrixForm