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SandBoxMatrixExample

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Editor: Bill Page Time: 2009/10/16 12:10:59 GMT-7
Note: clean up

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Matrix Example

\begin{axiom}
m1 := matrix([ [1, x/y, 0], [0, 1, 0], [0, 0, 1] ])
m2 := matrix([ [1, 0, 0], [0, 1, y/x], [0, 0, 1] ])
m3 := matrix([ [-1/y, 0, 0], [0, y/(x*z), 0], [0, 0, x/(x*y-z*y)] ])
m4 := matrix([ [1, 0, 0], [0, 1, 0], [0, y/x, 1] ])
m5 := matrix([ [1, 0, 0], [z/y, 1, 0], [0, 0, 1] ])
m6 := matrix([ [0, 0, 1], [0, 1, 0], [1, 0, 0] ])

T := m1*m2*m3*m4*m5*m6
M := matrix([ [0, -z, y], [z, 0, -x], [-y, x, 0] ])
R := matrix([ [r, 0, 0], [0, r, 0], [0, 0, r] ])

T - R
determinant(%)

T * vector([bx, by, bc])

\end{axiom}

\begin{axiom}
R := matrix([ [cos(t), -sin(t), 0], [sin(t), cos(t), 0], [0,0,1] ])
T := transpose(R)

e11 := matrix( [ [1,0,0], [0,0,0], [0,0,0] ] )
e12 := matrix( [ [0,1,0], [0,0,0], [0,0,0] ] )
e13 := matrix( [ [0,0,1], [0,0,0], [0,0,0] ] )
e21 := matrix( [ [0,0,0], [1,0,0], [0,0,0] ] )
e22 := matrix( [ [0,0,0], [0,1,0], [0,0,0] ] )
e23 := matrix( [ [0,0,0], [0,0,1], [0,0,0] ] )
e31 := matrix( [ [0,0,0], [0,0,0], [1,0,0] ] )
e32 := matrix( [ [0,0,0], [0,0,0], [0,1,0] ] )
e33 := matrix( [ [0,0,0], [0,0,0], [0,0,1] ] )

T*e11*R
T*e12*R
T*e13*R
T*e21*R
T*e22*R
T*e23*R
T*e31*R
T*e32*R
T*e33*R
\end{axiom}

Matrix Example

fricas

(1) -> m1 := matrix([ [1, x/y, 0], [0, 1, 0], [0, 0, 1] ])

$\label{eq1}\left[ \begin{array}{ccc} 1 &{\frac{x}{y}}& 0 \ 0 & 1 & 0 \ 0 & 0 & 1$

(1)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

m2 := matrix([ [1, 0, 0], [0, 1, y/x], [0, 0, 1] ])

$\label{eq2}\left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 &{\frac{y}{x}} \ 0 & 0 & 1$

(2)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

m3 := matrix([ [-1/y, 0, 0], [0, y/(x*z), 0], [0, 0, x/(x*y-z*y)] ])

$\label{eq3}\left[ \begin{array}{ccc} -{\frac{1}{y}}& 0 & 0 \ 0 &{\frac{y}{x \ z}}& 0 \ 0 & 0 & -{\frac{x}{{y \ z}-{x \ y}}}$

(3)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

m4 := matrix([ [1, 0, 0], [0, 1, 0], [0, y/x, 1] ])

$\label{eq4}\left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 &{\frac{y}{x}}& 1$

(4)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

m5 := matrix([ [1, 0, 0], [z/y, 1, 0], [0, 0, 1] ])

$\label{eq5}\left[ \begin{array}{ccc} 1 & 0 & 0 \ {\frac{z}{y}}& 1 & 0 \ 0 & 0 & 1$

(5)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

m6 := matrix([ [0, 0, 1], [0, 1, 0], [1, 0, 0] ])

$\label{eq6}\left[ \begin{array}{ccc} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0$

(6)

Type: Matrix(NonNegativeInteger?)

fricas

T := m1*m2*m3*m4*m5*m6

$\label{eq7}\left[ \begin{array}{ccc} -{\frac{x}{{y \ z}-{x \ y}}}& -{\frac{x}{{{z}^{2}}-{x \ z}}}& -{\frac{z}{{y \ z}-{x \ y}}} \ -{\frac{1}{z - x}}& -{\frac{y}{{{z}^{2}}-{x \ z}}}& -{\frac{1}{z - x}} \ -{\frac{x}{{y \ z}-{x \ y}}}& -{\frac{1}{z - x}}& -{\frac{z}{{y \ z}-{x \ y}}}$

(7)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

M := matrix([ [0, -z, y], [z, 0, -x], [-y, x, 0] ])

$\label{eq8}\left[ \begin{array}{ccc} 0 & - z & y \ z & 0 & - x \ - y & x & 0$

(8)

Type: Matrix(Polynomial(Integer))

fricas

R := matrix([ [r, 0, 0], [0, r, 0], [0, 0, r] ])

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

(9)

Type: Matrix(Polynomial(Integer))

fricas

T - R

$\label{eq10}\left[ \begin{array}{ccc} {\frac{-{r \ y \ z}+{r \ x \ y}- x}{{y \ z}-{x \ y}}}& -{\frac{x}{{{z}^{2}}-{x \ z}}}& -{\frac{z}{{y \ z}-{x \ y}}} \ -{\frac{1}{z - x}}&{\frac{-{r \ {{z}^{2}}}+{r \ x \ z}- y}{{{z}^{2}}-{x \ z}}}& -{\frac{1}{z - x}} \ -{\frac{x}{{y \ z}-{x \ y}}}& -{\frac{1}{z - x}}&{\frac{{{\left(-{r \ y}- 1 \right)}\ z}+{r \ x \ y}}{{y \ z}-{x \ y}}}$

(10)

Type: Matrix(Fraction(Polynomial(Integer)))

fricas

determinant(%)

$\label{eq11}\frac{{{\left(-{{{r}^{3}}\ y}-{{r}^{2}}\right)}\ {{z}^{2}}}+{{\left({{{r}^{3}}\ x \ y}-{{{r}^{2}}\ x}\right)}\ z}-{{{r}^{2}}\ {{y}^{2}}}- 1}{{y \ {{z}^{2}}}-{x \ y \ z}}$

(11)

Type: Fraction(Polynomial(Integer))

fricas

T * vector([bx, by, bc])

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{\frac{-{bc \ {{z}^{2}}}-{bx \ x \ z}-{ \hbox{ by } \ x \ y}}{{y \ {{z}^{2}}}-{x \ y \ z}}}, \: \right. \ \ \displaystyle \left.{\frac{{{\left(- bx - bc \right)}\ z}-{ \hbox{ by } \ y}}{{{z}^{2}}-{x \ z}}}, \: \right. \ \ \displaystyle \left.{\frac{-{bc \ z}-{ \hbox{ by } \ y}-{bx \ x}}{{y \ z}-{x \ y}}}\right]$

(12)

Type: Vector(Fraction(Polynomial(Integer)))

fricas

R := matrix([ [cos(t), -sin(t), 0], [sin(t), cos(t), 0], [0,0,1] ])

$\label{eq13}\left[ \begin{array}{ccc} {\cos \left({t}\right)}& -{\sin \left({t}\right)}& 0 \ {\sin \left({t}\right)}&{\cos \left({t}\right)}& 0 \ 0 & 0 & 1$

(13)

Type: Matrix(Expression(Integer))

fricas

T := transpose(R)

$\label{eq14}\left[ \begin{array}{ccc} {\cos \left({t}\right)}&{\sin \left({t}\right)}& 0 \ -{\sin \left({t}\right)}&{\cos \left({t}\right)}& 0 \ 0 & 0 & 1$

(14)

Type: Matrix(Expression(Integer))

fricas

e11 := matrix( [ [1,0,0], [0,0,0], [0,0,0] ] )

$\label{eq15}\left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0$

(15)

Type: Matrix(NonNegativeInteger?)

fricas

e12 := matrix( [ [0,1,0], [0,0,0], [0,0,0] ] )

$\label{eq16}\left[ \begin{array}{ccc} 0 & 1 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0$

(16)

Type: Matrix(NonNegativeInteger?)

fricas

e13 := matrix( [ [0,0,1], [0,0,0], [0,0,0] ] )

$\label{eq17}\left[ \begin{array}{ccc} 0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0$

(17)

Type: Matrix(NonNegativeInteger?)

fricas

e21 := matrix( [ [0,0,0], [1,0,0], [0,0,0] ] )

$\label{eq18}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0$

(18)

Type: Matrix(NonNegativeInteger?)

fricas

e22 := matrix( [ [0,0,0], [0,1,0], [0,0,0] ] )

$\label{eq19}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0$

(19)

Type: Matrix(NonNegativeInteger?)

fricas

e23 := matrix( [ [0,0,0], [0,0,1], [0,0,0] ] )

$\label{eq20}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 1 \ 0 & 0 & 0$

(20)

Type: Matrix(NonNegativeInteger?)

fricas

e31 := matrix( [ [0,0,0], [0,0,0], [1,0,0] ] )

$\label{eq21}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 1 & 0 & 0$

(21)

Type: Matrix(NonNegativeInteger?)

fricas

e32 := matrix( [ [0,0,0], [0,0,0], [0,1,0] ] )

$\label{eq22}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 1 & 0$

(22)

Type: Matrix(NonNegativeInteger?)

fricas

e33 := matrix( [ [0,0,0], [0,0,0], [0,0,1] ] )

$\label{eq23}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1$

(23)

Type: Matrix(NonNegativeInteger?)

fricas

T*e11*R

$\label{eq24}\left[ \begin{array}{ccc} {{\cos \left({t}\right)}^{2}}& -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 \ -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}&{{\sin \left({t}\right)}^{2}}& 0 \ 0 & 0 & 0$

(24)

Type: Matrix(Expression(Integer))

fricas

T*e12*R

$\label{eq25}\left[ \begin{array}{ccc} {{\cos \left({t}\right)}\ {\sin \left({t}\right)}}&{{\cos \left({t}\right)}^{2}}& 0 \ -{{\sin \left({t}\right)}^{2}}& -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 \ 0 & 0 & 0$

(25)

Type: Matrix(Expression(Integer))

fricas

T*e13*R

$\label{eq26}\left[ \begin{array}{ccc} 0 & 0 &{\cos \left({t}\right)} \ 0 & 0 & -{\sin \left({t}\right)} \ 0 & 0 & 0$

(26)

Type: Matrix(Expression(Integer))

fricas

T*e21*R

$\label{eq27}\left[ \begin{array}{ccc} {{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& -{{\sin \left({t}\right)}^{2}}& 0 \ {{\cos \left({t}\right)}^{2}}& -{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 \ 0 & 0 & 0$

(27)

Type: Matrix(Expression(Integer))

fricas

T*e22*R

$\label{eq28}\left[ \begin{array}{ccc} {{\sin \left({t}\right)}^{2}}&{{\cos \left({t}\right)}\ {\sin \left({t}\right)}}& 0 \ {{\cos \left({t}\right)}\ {\sin \left({t}\right)}}&{{\cos \left({t}\right)}^{2}}& 0 \ 0 & 0 & 0$