Boas : Ch3 Sec 11 MOD

Clear["Global`*"]

Follow example

mat={{3,-1},{-1,3}};
Eigensystem[mat]

mat //MatrixForm

lam = λ DiagonalMatrix[{1,1}];
lam //MatrixForm

mat - lam //MatrixForm

eq0= mat - lam //Det

Solve[eq0==0,λ]

Find Eigenvectors: λ ={2,4}

vec={x,y};
eq1= mat.vec == λ vec

eq2= eq1 //Thread

Solve[ eq2 /.{λ->2},{x,y}]

eq3= Join[eq2,{ (x^2)+(y^2)==1}]

Solve[eq3 /.{λ->2},{x,y}]

Solve[eq3 /.{λ->4},{x,y}]

Examine

eval=Eigenvalues[mat]

evec=Eigenvectors[mat]

mat.evec[[1]] //Simplify

mat.evec[[1]] == 4 evec[[1]]//Simplify

mat.evec[[2]] //Simplify

mat.evec[[2]] == 2 evec[[2]]//Simplify

evec=Normalize /@ evec

Transpose[evec].mat.evec  //Simplify//MatrixForm

Boas : Ch3 Sec 11

Clear["Global`*"]

Follow example

mat={{5,-2},{-2,2}};
mat //MatrixForm

lam = λ DiagonalMatrix[{1,1}];
lam //MatrixForm

mat - lam //MatrixForm

eq0= mat - lam //Det

Solve[eq0==0,λ]

Find Eigenvectors: λ ={6,1}

vec={x,y};
eq1= mat.vec == λ vec

eq2= eq1 //Thread

Solve[ eq2 /.{λ->6},{x,y}]

eq3= Join[eq2,{ (x^2)+(y^2)==1}]

Solve[eq3 /.{λ->6},{x,y}]

Solve[eq3 /.{λ->1},{x,y}]

Examine

eval=Eigenvalues[mat]

evec=Eigenvectors[mat]

mat.evec[[1]] //Simplify

mat.evec[[2]] //Simplify

evec=Normalize /@ evec

Transpose[evec].mat.evec  //Simplify//MatrixForm

Boas : Ch3 Example 3 p.165

Clear["Global`*"]

Follow example

pot= (1/2)k (x^2) +(1/2)k (((x-y))^2) +(1/2)k (y^2) //Simplify

D[pot,x]
D[pot,y]

mat={{2,-1},{-1,2}};
mat //MatrixForm

lam = λ DiagonalMatrix[{1,1}];
lam //MatrixForm

mat - lam //MatrixForm

eq0= mat - lam //Det

Solve[eq0==0,λ]

Find Eigenvectors: λ ={1,3}

vec={x,y};
eq1= mat.vec == λ vec

eq2= eq1 //Thread

Solve[ eq2 /.{λ->1},{x,y}]

eq3= Join[eq2,{ (x^2)+(y^2)==1}]

Solve[eq3 /.{λ->1},{x,y}]

Solve[eq3 /.{λ->3},{x,y}]

Examine

eval=Eigenvalues[mat]

evec=Eigenvectors[mat]

mat.evec[[1]] //Simplify

mat.evec[[1]] == 3 evec[[1]]//Simplify

mat.evec[[2]] //Simplify

mat.evec[[2]] == 1 evec[[2]]//Simplify

evec=Normalize /@ evec

Transpose[evec].mat.evec  //Simplify//MatrixForm

Change coordinates

eq1={ z1==((x+y)/(Sqrt[2])),  z2==((x-y)/(Sqrt[2]))}

sol= Solve[eq1,{x,y}][[1]] //Simplify

pot

pot /.sol //Simplify