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Symbolic Matrices

fricas
A:=matrix [[x,y],[z,w]]

\label{eq1}\left[ 
\begin{array}{cc}
x & y 
\
z & w 
(1)
Type: Matrix(Polynomial(Integer))
fricas
A+1

\label{eq2}\left[ 
\begin{array}{cc}
{x + 1}& y 
\
z &{w + 1}
(2)
Type: SquareMatrix?(2,Polynomial(Integer))

fricas
A+2

\label{eq3}\left[ 
\begin{array}{cc}
{x + 2}& y 
\
z &{w + 2}
(3)
Type: SquareMatrix?(2,Polynomial(Integer))


Use the Edit and Preview Functions

Hey, why not learn to use the edit function instead of entering such a large number of similar comments?

Look at the top right hand side of the page.


fricas
N:=matrix[[0],[0]]

\label{eq4}\left[ 
\begin{array}{c}
0 
\
0 
(4)
Type: Matrix(NonNegativeInteger)
fricas
L:=[[sqrt(-1)*sin(x)+cos(x)],[-sqrt(-1)*sin(x)+cos(x)]]

\label{eq5}\left[{\left[{{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}, \:{\left[{-{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}\right](5)
Type: List(List(Expression(Integer)))
fricas
A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]

\label{eq6}\left[ 
\begin{array}{cc}
{\cos \left({x}\right)}& -{\sin \left({x}\right)}
\
{\sin \left({x}\right)}&{\cos \left({x}\right)}
(6)
Type: Matrix(Expression(Integer))
fricas
v:=matrix[[v11],[v12]]

\label{eq7}\left[ 
\begin{array}{c}
v 11 
\
v 12 
(7)
Type: Matrix(Polynomial(Integer))
fricas
C:=A*v-L(1,1)*v

\label{eq8}\left[ 
\begin{array}{c}
{{\left(-{v 11 \ {\sqrt{- 1}}}- v 12 \right)}\ {\sin \left({x}\right)}}
\
{{\left(-{v 12 \ {\sqrt{- 1}}}+ v 11 \right)}\ {\sin \left({x}\right)}}
(8)
Type: Matrix(Expression(Integer))
fricas
solve(C(1,1)=0,v11)

\label{eq9}\left[{v 11 = -{v 12 \over{\sqrt{- 1}}}}\right](9)
Type: List(Equation(Expression(Integer)))
fricas
solve(C(2,1)=0,v12)

\label{eq10}\left[{v 12 ={v 11 \over{\sqrt{- 1}}}}\right](10)
Type: List(Equation(Expression(Integer)))
fricas
V:=matrix[[1/sqrt(-1),1],[1,-1/sqrt(-1)]]

\label{eq11}\left[ 
\begin{array}{cc}
-{\sqrt{- 1}}& 1 
\
1 &{\sqrt{- 1}}
(11)
Type: Matrix(AlgebraicNumber?)
fricas
Z:=matrix[[V(2,2),-V(1,2)],[-V(2,1),V(1,1)]]

\label{eq12}\left[ 
\begin{array}{cc}
{\sqrt{- 1}}& - 1 
\
- 1 & -{\sqrt{- 1}}
(12)
Type: Matrix(AlgebraicNumber?)
fricas
W:=(V(1,1)*V(2,2) - V(1,2)*V(2,1))

\label{eq13}0(13)
Type: AlgebraicNumber?

fricas
N:=matrix[[0],[0]]

\label{eq14}\left[ 
\begin{array}{c}
0 
\
0 
(14)
Type: Matrix(NonNegativeInteger)
fricas
L:=[[sqrt(-1)*sin(x)+cos(x)],[-sqrt(-1)*sin(x)+cos(x)]]

\label{eq15}\left[{\left[{{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}, \:{\left[{-{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}\right](15)
Type: List(List(Expression(Integer)))
fricas
A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]

\label{eq16}\left[ 
\begin{array}{cc}
{\cos \left({x}\right)}& -{\sin \left({x}\right)}
\
{\sin \left({x}\right)}&{\cos \left({x}\right)}
(16)
Type: Matrix(Expression(Integer))
fricas
v:=matrix[[v11],[v12]]

\label{eq17}\left[ 
\begin{array}{c}
v 11 
\
v 12 
(17)
Type: Matrix(Polynomial(Integer))
fricas
C:=A*v-L(1,1)*v

\label{eq18}\left[ 
\begin{array}{c}
{{\left(-{v 11 \ {\sqrt{- 1}}}- v 12 \right)}\ {\sin \left({x}\right)}}
\
{{\left(-{v 12 \ {\sqrt{- 1}}}+ v 11 \right)}\ {\sin \left({x}\right)}}
(18)
Type: Matrix(Expression(Integer))
fricas
solve(C(1,1)=0,v11)

\label{eq19}\left[{v 11 = -{v 12 \over{\sqrt{- 1}}}}\right](19)
Type: List(Equation(Expression(Integer)))
fricas
solve(C(2,1)=0,v12)

\label{eq20}\left[{v 12 ={v 11 \over{\sqrt{- 1}}}}\right](20)
Type: List(Equation(Expression(Integer)))
fricas
V:=matrix[[1/sqrt(-1),1],[1,-1/sqrt(-1)]]

\label{eq21}\left[ 
\begin{array}{cc}
-{\sqrt{- 1}}& 1 
\
1 &{\sqrt{- 1}}
(21)
Type: Matrix(AlgebraicNumber?)
fricas
Z:=matrix[[V(2,2),-V(1,2)],[-V(2,1),V(1,1)]]

\label{eq22}\left[ 
\begin{array}{cc}
{\sqrt{- 1}}& - 1 
\
- 1 & -{\sqrt{- 1}}
(22)
Type: Matrix(AlgebraicNumber?)
fricas
V(1,1)*V(2,2)

\label{eq23}1(23)
Type: AlgebraicNumber?
fricas
V(1,2)*V(2,1)

\label{eq24}1(24)
Type: AlgebraicNumber?

fricas
N:=matrix[[0],[0]]

\label{eq25}\left[ 
\begin{array}{c}
0 
\
0 
(25)
Type: Matrix(NonNegativeInteger)
fricas
L:=[[sqrt(-1)*sin(x)+cos(x)],[-sqrt(-1)*sin(x)+cos(x)]]

\label{eq26}\left[{\left[{{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}, \:{\left[{-{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}\right](26)
Type: List(List(Expression(Integer)))
fricas
A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]

\label{eq27}\left[ 
\begin{array}{cc}
{\cos \left({x}\right)}& -{\sin \left({x}\right)}
\
{\sin \left({x}\right)}&{\cos \left({x}\right)}
(27)
Type: Matrix(Expression(Integer))
fricas
v:=matrix[[v11],[v12]]

\label{eq28}\left[ 
\begin{array}{c}
v 11 
\
v 12 
(28)
Type: Matrix(Polynomial(Integer))
fricas
C:=A*v-L(1,1)*v

\label{eq29}\left[ 
\begin{array}{c}
{{\left(-{v 11 \ {\sqrt{- 1}}}- v 12 \right)}\ {\sin \left({x}\right)}}
\
{{\left(-{v 12 \ {\sqrt{- 1}}}+ v 11 \right)}\ {\sin \left({x}\right)}}
(29)
Type: Matrix(Expression(Integer))
fricas
solve(C(1,1)=0,v11)

\label{eq30}\left[{v 11 = -{v 12 \over{\sqrt{- 1}}}}\right](30)
Type: List(Equation(Expression(Integer)))
fricas
solve(C(2,1)=0,v12)

\label{eq31}\left[{v 12 ={v 11 \over{\sqrt{- 1}}}}\right](31)
Type: List(Equation(Expression(Integer)))
fricas
T:=matrix[[1/sqrt(-1),1],[1,-1/sqrt(-1)]]

\label{eq32}\left[ 
\begin{array}{cc}
-{\sqrt{- 1}}& 1 
\
1 &{\sqrt{- 1}}
(32)
Type: Matrix(AlgebraicNumber?)
fricas
a:=sqrt(T(1,1)^2+T(2,1)^2)

\label{eq33}0(33)
Type: AlgebraicNumber?
fricas
b=sqrt(T(1,2)^2+T(2,2)^2)

\label{eq34}b = 0(34)
Type: Equation(Polynomial(AlgebraicNumber?))
fricas
Z:=matrix[[V(2,2),-V(1,2)],[-V(2,1),V(1,1)]]

\label{eq35}\left[ 
\begin{array}{cc}
{\sqrt{- 1}}& - 1 
\
- 1 & -{\sqrt{- 1}}
(35)
Type: Matrix(AlgebraicNumber?)
fricas
V(1,1)*V(2,2)

\label{eq36}1(36)
Type: AlgebraicNumber?
fricas
V(1,2)*V(2,1)

\label{eq37}1(37)
Type: AlgebraicNumber?

fricas
N:=matrix[[0],[0]]

\label{eq38}\left[ 
\begin{array}{c}
0 
\
0 
(38)
Type: Matrix(NonNegativeInteger)
fricas
L:=[[sqrt(-1)*sin(x)+cos(x)],[-sqrt(-1)*sin(x)+cos(x)]]

\label{eq39}\left[{\left[{{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}, \:{\left[{-{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}\right]}\right](39)
Type: List(List(Expression(Integer)))
fricas
A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]

\label{eq40}\left[ 
\begin{array}{cc}
{\cos \left({x}\right)}& -{\sin \left({x}\right)}
\
{\sin \left({x}\right)}&{\cos \left({x}\right)}
(40)
Type: Matrix(Expression(Integer))
fricas
v:=matrix[[v11],[v12]]

\label{eq41}\left[ 
\begin{array}{c}
v 11 
\
v 12 
(41)
Type: Matrix(Polynomial(Integer))
fricas
C:=A*v-L(1,1)*v

\label{eq42}\left[ 
\begin{array}{c}
{{\left(-{v 11 \ {\sqrt{- 1}}}- v 12 \right)}\ {\sin \left({x}\right)}}
\
{{\left(-{v 12 \ {\sqrt{- 1}}}+ v 11 \right)}\ {\sin \left({x}\right)}}
(42)
Type: Matrix(Expression(Integer))
fricas
solve(C(1,1)=0,v11)

\label{eq43}\left[{v 11 = -{v 12 \over{\sqrt{- 1}}}}\right](43)
Type: List(Equation(Expression(Integer)))
fricas
solve(C(2,1)=0,v12)

\label{eq44}\left[{v 12 ={v 11 \over{\sqrt{- 1}}}}\right](44)
Type: List(Equation(Expression(Integer)))
fricas
T:=matrix[[1/sqrt(-1),1],[1,-1/sqrt(-1)]]

\label{eq45}\left[ 
\begin{array}{cc}
-{\sqrt{- 1}}& 1 
\
1 &{\sqrt{- 1}}
(45)
Type: Matrix(AlgebraicNumber?)
fricas
sqrt(T(1,1)^2+T(2,1)^2)

\label{eq46}0(46)
Type: AlgebraicNumber?
fricas
sqrt(T(1,2)^2+T(2,2)^2)

\label{eq47}0(47)
Type: AlgebraicNumber?
fricas
Z:=matrix[[V(2,2),-V(1,2)],[-V(2,1),V(1,1)]]

\label{eq48}\left[ 
\begin{array}{cc}
{\sqrt{- 1}}& - 1 
\
- 1 & -{\sqrt{- 1}}
(48)
Type: Matrix(AlgebraicNumber?)
fricas
V(1,1)*V(2,2)

\label{eq49}1(49)
Type: AlgebraicNumber?
fricas
V(1,2)*V(2,1)

\label{eq50}1(50)
Type: AlgebraicNumber?

fricas
)clear all
All user variables and function definitions have been cleared. B := %i*sqrt(a^2 + b^2 + c^2)

\label{eq51}i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}(51)
Type: Expression(Complex(Integer))
fricas
A := matrix[ [B, c, -b], [-c, B, a], [b, -a, B] ]

\label{eq52}\left[ 
\begin{array}{ccc}
{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}& c & - b 
\
- c &{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}& a 
\
b & - a &{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}
(52)
Type: Matrix(Expression(Complex(Integer)))
fricas
rowEchelon(A)

\label{eq53}\left[ 
\begin{array}{ccc}
1 & 0 &{{{a \  c \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}+{i \  b \ {{c}^{2}}}+{i \ {{b}^{3}}}+{i \ {{a}^{2}}\  b}}\over{{\left({{b}^{2}}+{{a}^{2}}\right)}\ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}}
\
0 & 1 &{{-{i \  a \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}+{b \  c}}\over{{{b}^{2}}+{{a}^{2}}}}
\
0 & 0 & 0 
(53)
Type: Matrix(Expression(Complex(Integer)))
fricas
B := -%i*sqrt(a^2 + b^2 + c^2)

\label{eq54}-{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}(54)
Type: Expression(Complex(Integer))
fricas
A := matrix[ [B, c, -b], [-c, B, a], [b, -a, B] ]

\label{eq55}\left[ 
\begin{array}{ccc}
-{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}& c & - b 
\
- c & -{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}& a 
\
b & - a & -{i \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}
(55)
Type: Matrix(Expression(Complex(Integer)))
fricas
rowEchelon(A)

\label{eq56}\left[ 
\begin{array}{ccc}
1 & 0 &{{{a \  c \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}-{i \  b \ {{c}^{2}}}-{i \ {{b}^{3}}}-{i \ {{a}^{2}}\  b}}\over{{\left({{b}^{2}}+{{a}^{2}}\right)}\ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}}
\
0 & 1 &{{{i \  a \ {\sqrt{{{c}^{2}}+{{b}^{2}}+{{a}^{2}}}}}+{b \  c}}\over{{{b}^{2}}+{{a}^{2}}}}
\
0 & 0 & 0 
(56)
Type: Matrix(Expression(Complex(Integer)))

Write:
  \begin{axiom}
  M := matrix [[1,1,1,1],[a,a,b,b],[1,1,1,1],[a,a,b,b]]
  \end{axiom}

fricas
M := matrix [[1,1,1,1],[a,a,b,b],[1,1,1,1],[a,a,b,b]]

\label{eq57}\left[ 
\begin{array}{cccc}
1 & 1 & 1 & 1 
\
a & a & b & b 
\
1 & 1 & 1 & 1 
\
a & a & b & b 
(57)
Type: Matrix(Polynomial(Integer))




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