MathAction SandBoxREN

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fricas

As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])

$\label{eq1}\left[ \begin{array}{cccc} - 3 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1$

(1)

Type: Matrix(Integer)

fricas

A := subMatrix(As, 2,4,2,4)

$\label{eq2}\left[ \begin{array}{ccc} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1$

(2)

Type: Matrix(Integer)

fricas

ob := orthonormalBasis(A)

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{c} -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}} \ -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}} \ {{\sqrt{2}}\over{\sqrt{3}}}$

(3)

Type: List(Matrix(Expression(Integer)))

fricas

P : Matrix(Expression Integer) := new(3,3,0)

$\label{eq4}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0$

(4)

Type: Matrix(Expression(Integer))

fricas

setsubMatrix!(P,1,1,ob.3)

$\label{eq5}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}& 0 & 0 \ {1 \over{\sqrt{3}}}& 0 & 0 \ {1 \over{\sqrt{3}}}& 0 & 0$

(5)

Type: Matrix(Expression(Integer))

fricas

setsubMatrix!(P,1,2,ob.1)

$\label{eq6}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& 0 \ {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& 0 \ {1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0$

(6)

Type: Matrix(Expression(Integer))

fricas

setsubMatrix!(P,1,3,ob.2)

$\label{eq7}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}} \ {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}} \ {1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0$

(7)

Type: Matrix(Expression(Integer))

fricas

Pt := transpose(P)

$\label{eq8}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}} \ -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}} \ -{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0$

(8)

Type: Matrix(Expression(Integer))

fricas

Ps : Matrix(Expression Integer) := new(4,4,0)

$\label{eq9}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(9)

Type: Matrix(Expression(Integer))

fricas

Ps(1,1) := 1

$\label{eq10}1$

(10)

Type: Expression(Integer)

fricas

setsubMatrix!(Ps,2,2,P)

$\label{eq11}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 &{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}} \ 0 &{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}} \ 0 &{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0$

(11)

Type: Matrix(Expression(Integer))

fricas

PsT := transpose(Ps)

$\label{eq12}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 &{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}} \ 0 & -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}} \ 0 & -{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0$

(12)

Type: Matrix(Expression(Integer))

fricas

PsTAsPs := PsT * As * Ps

$\label{eq13}\left[ \begin{array}{cccc} - 3 &{3 \over{\sqrt{3}}}& 0 & 0 \ {3 \over{\sqrt{3}}}& 3 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(13)

Type: Matrix(Expression(Integer))

fricas

b1 := PsTAsPs(2,1)

$\label{eq14}3 \over{\sqrt{3}}$

(14)

Type: Expression(Integer)

fricas

l1 := PsTAsPs(2,2)

$\label{eq15}3$

(15)

Type: Expression(Integer)

fricas

Us : Matrix(Expression Integer) := new(4,4,0)

$\label{eq16}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(16)

Type: Matrix(Expression(Integer))

fricas

Us(1,1) := 1

$\label{eq17}1$

(17)

Type: Expression(Integer)

fricas

Us(2,2) := 1

$\label{eq18}1$

(18)

Type: Expression(Integer)

fricas

Us(3,3) := 1

$\label{eq19}1$

(19)

Type: Expression(Integer)

fricas

Us(4,4) := 1

$\label{eq20}1$

(20)

Type: Expression(Integer)

fricas

Us(2,1) := -b1 / l1

$\label{eq21}-{1 \over{\sqrt{3}}}$

(21)

Type: Expression(Integer)

fricas

Us

$\label{eq22}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ -{1 \over{\sqrt{3}}}& 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(22)

Type: Matrix(Expression(Integer))

fricas

PsUs := Ps * Us

$\label{eq23}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ -{1 \over 3}&{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}} \ -{1 \over 3}&{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}} \ -{1 \over 3}&{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0$

(23)

Type: Matrix(Expression(Integer))

fricas

PsUsT := transpose(PsUs)

$\label{eq24}\left[ \begin{array}{cccc} 1 & -{1 \over 3}& -{1 \over 3}& -{1 \over 3} \ 0 &{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}} \ 0 & -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}} \ 0 & -{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0$

(24)

Type: Matrix(Expression(Integer))

fricas

PsUsTAsPsUs := PsUsT * As * PsUs

$\label{eq25}\left[ \begin{array}{cccc} - 4 & 0 & 0 & 0 \ 0 & 3 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(25)

Type: Matrix(Expression(Integer))

fricas

C := inverse(PsUs)

$\label{eq26}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ {{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3} \ 0 & -{{\sqrt{3}}\over{3 \ {\sqrt{2}}}}& -{{{\sqrt{2}}\ {\sqrt{3}}}\over 6}&{{2 \ {\sqrt{3}}}\over{3 \ {\sqrt{2}}}} \ 0 & -{{\sqrt{2}}\over 2}&{{\sqrt{2}}\over 2}& 0$

(26)

Type: Union(Matrix(Expression(Integer)),...)

fricas

c := PsUsTAsPsUs(1,1)

$\label{eq27}- 4$

(27)

Type: Expression(Integer)

fricas

gQ := PsUsTAsPsUs / c

$\label{eq28}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & -{3 \over 4}& 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(28)

Type: Matrix(Expression(Integer))

fricas

x1 := transpose(matrix([[1,2,3,4]]))

$\label{eq29}\left[ \begin{array}{c} 1 \ 2 \ 3 \ 4$

(29)

Type: Matrix(Integer)

fricas

v1 := transpose(x1) * As * x1

$\label{eq30}\left[ \begin{array}{c} {96}$

(30)

Type: Matrix(Integer)

fricas

x2 := C * x1

$\label{eq31}\left[ \begin{array}{c} 1 \ {{{10}\ {\sqrt{3}}}\over 3} \ {{\sqrt{3}}\over{\sqrt{2}}} \ {{\sqrt{2}}\over 2}$

(31)

Type: Matrix(Expression(Integer))

fricas

v2 := transpose(x2) * PsUsTAsPsUs * x2

$\label{eq32}\left[ \begin{array}{c} {96}$

(32)

Type: Matrix(Expression(Integer))

... --ren, Fri, 07 Jul 2006 14:33:03 -0500 reply

fricas

)clear value c

As := matrix([ [-c,-1,-2,-1], [-1,3,-1,0], [-2,-1,3,0], [-1,0,0,-6]])

$\label{eq33}\left[ \begin{array}{cccc} - c & - 1 & - 2 & - 1 \ - 1 & 3 & - 1 & 0 \ - 2 & - 1 & 3 & 0 \ - 1 & 0 & 0 & - 6$

(33)

Type: Matrix(Polynomial(Integer))

fricas

A := subMatrix(As, 2,4,2,4)

$\label{eq34}\left[ \begin{array}{ccc} 3 & - 1 & 0 \ - 1 & 3 & 0 \ 0 & 0 & - 6$

(34)

Type: Matrix(Polynomial(Integer))

fricas

ob := orthonormalBasis(A)

$\label{eq35}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{c} 0 \ 0 \ 1$

(35)

Type: List(Matrix(Expression(Integer)))

fricas

P : Matrix(Expression Integer) := new(3,3,0)

$\label{eq36}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0$

(36)

Type: Matrix(Expression(Integer))

fricas

setsubMatrix!(P,1,1,ob.3)

$\label{eq37}\left[ \begin{array}{ccc} -{1 \over{\sqrt{2}}}& 0 & 0 \ {1 \over{\sqrt{2}}}& 0 & 0 \ 0 & 0 & 0$

(37)

Type: Matrix(Expression(Integer))

fricas

setsubMatrix!(P,1,2,ob.1)

$\label{eq38}\left[ \begin{array}{ccc} -{1 \over{\sqrt{2}}}& 0 & 0 \ {1 \over{\sqrt{2}}}& 0 & 0 \ 0 & 1 & 0$

(38)

Type: Matrix(Expression(Integer))

fricas

setsubMatrix!(P,1,3,ob.2)

$\label{eq39}\left[ \begin{array}{ccc} -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}} \ {1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}} \ 0 & 1 & 0$

(39)

Type: Matrix(Expression(Integer))

fricas

Pt := transpose(P)

$\label{eq40}\left[ \begin{array}{ccc} -{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0 \ 0 & 0 & 1 \ {1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0$

(40)

Type: Matrix(Expression(Integer))

fricas

Ps : Matrix(Expression Integer) := new(4,4,0)

$\label{eq41}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(41)

Type: Matrix(Expression(Integer))

fricas

Ps(1,1) := 1

$\label{eq42}1$

(42)

Type: Expression(Integer)

fricas

setsubMatrix!(Ps,2,2,P)

$\label{eq43}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}} \ 0 &{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}} \ 0 & 0 & 1 & 0$

(43)

Type: Matrix(Expression(Integer))

fricas

PsT := transpose(Ps)

$\label{eq44}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & -{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0 \ 0 & 0 & 0 & 1 \ 0 &{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0$

(44)

Type: Matrix(Expression(Integer))

fricas

PsTAsPs := PsT * As * Ps

$\label{eq45}\left[ \begin{array}{cccc} - c & -{1 \over{\sqrt{2}}}& - 1 & -{3 \over{\sqrt{2}}} \ -{1 \over{\sqrt{2}}}& 4 & 0 & 0 \ - 1 & 0 & - 6 & 0 \ -{3 \over{\sqrt{2}}}& 0 & 0 & 2$

(45)

Type: Matrix(Expression(Integer))

fricas

Us : Matrix(Expression Integer) := new(4,4,0)

$\label{eq46}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(46)

Type: Matrix(Expression(Integer))

fricas

Us(1,1) := 1

$\label{eq47}1$

(47)

Type: Expression(Integer)

fricas

Us(2,2) := 1

$\label{eq48}1$

(48)

Type: Expression(Integer)

fricas

Us(3,3) := 1

$\label{eq49}1$

(49)

Type: Expression(Integer)

fricas

Us(4,4) := 1

$\label{eq50}1$

(50)

Type: Expression(Integer)

fricas

Us(2,1) := -PsTAsPs(2,1) / PsTAsPs(2,2)

$\label{eq51}1 \over{4 \ {\sqrt{2}}}$

(51)

Type: Expression(Integer)

fricas

Us(3,1) := -PsTAsPs(3,1) / PsTAsPs(3,3)

$\label{eq52}-{1 \over 6}$

(52)

Type: Expression(Integer)

fricas

Us(4,1) := -PsTAsPs(4,1) / PsTAsPs(4,4)

$\label{eq53}3 \over{2 \ {\sqrt{2}}}$

(53)

Type: Expression(Integer)

fricas

Us

$\label{eq54}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ {1 \over{4 \ {\sqrt{2}}}}& 1 & 0 & 0 \ -{1 \over 6}& 0 & 1 & 0 \ {3 \over{2 \ {\sqrt{2}}}}& 0 & 0 & 1$

(54)

Type: Matrix(Expression(Integer))

fricas

PsUs := Ps * Us

$\label{eq55}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ {5 \over 8}& -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}} \ {7 \over 8}&{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}} \ -{1 \over 6}& 0 & 1 & 0$

(55)

Type: Matrix(Expression(Integer))

fricas

PsUsT := transpose(PsUs)

$\label{eq56}\left[ \begin{array}{cccc} 1 &{5 \over 8}&{7 \over 8}& -{1 \over 6} \ 0 & -{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0 \ 0 & 0 & 0 & 1 \ 0 &{1 \over{\sqrt{2}}}&{1 \over{\sqrt{2}}}& 0$

(56)

Type: Matrix(Expression(Integer))

fricas

PsUsTAsPsUs := PsUsT * As * PsUs

$\label{eq57}\left[ \begin{array}{cccc} {{-{{24}\ c}-{53}}\over{24}}& 0 & 0 & 0 \ 0 & 4 & 0 & 0 \ 0 & 0 & - 6 & 0 \ 0 & 0 & 0 & 2$

(57)

Type: Matrix(Expression(Integer))

fricas

cc := PsUsTAsPsUs(1,1)

$\label{eq58}{-{{24}\ c}-{53}}\over{24}$

(58)

Type: Expression(Integer)

fricas

so := solve(cc = 0, c)

$\label{eq59}\left[{c = -{{53}\over{24}}}\right]$

(59)

Type: List(Equation(Expression(Integer)))

fricas

c0 := rhs so.1

$\label{eq60}-{{53}\over{24}}$

(60)

Type: Expression(Integer)

fricas

gQ := PsUsTAsPsUs / cc

$\label{eq61}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & -{{96}\over{{{24}\ c}+{53}}}& 0 & 0 \ 0 & 0 &{{144}\over{{{24}\ c}+{53}}}& 0 \ 0 & 0 & 0 & -{{48}\over{{{24}\ c}+{53}}}$

(61)

Type: Matrix(Expression(Integer))

fricas

eval(PsUsTAsPsUs, c = c0)

$\label{eq62}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 4 & 0 & 0 \ 0 & 0 & - 6 & 0 \ 0 & 0 & 0 & 2$

(62)

Type: Matrix(Expression(Integer))