from sympy import *
from IPython.display import display,Math
x,y,z, t,n = symbols('x y z t n')

Matrizes e Operações

display(
    
Matrix([
    [1,   6,9],
    [E*2, 0,0],
    [-5.5,2,x]
]),
    
Matrix([x,y,z])
    
)

$\displaystyle \left[\begin{matrix}1 & 6 & 9\\2 e & 0 & 0\\-5.5 & 2 & x\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}x\\y\\z\end{matrix}\right]$

---

Operações

A = Matrix([
    [1,2,3],
    [4,5,6],
    [3,8,2]
])

b = Matrix([3,10,12])

display(Math('A ='+latex(A)))
display(Math('b ='+latex(b)))
display(Math('A*b ='+latex(A*b)))
display(Math('3*b ='+latex(3*b)))
display(Math('A² ='+latex(A**2)))
display(Math('A^{-1} ='+latex(A**-1)))
display(Math('inv(A) ='+latex(A.inv())))

$\displaystyle A =\left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\3 & 8 & 2\end{matrix}\right]$

$\displaystyle b =\left[\begin{matrix}3\\10\\12\end{matrix}\right]$

$\displaystyle A*b =\left[\begin{matrix}59\\134\\113\end{matrix}\right]$

$\displaystyle 3*b =\left[\begin{matrix}9\\30\\36\end{matrix}\right]$

$\displaystyle A² =\left[\begin{matrix}18 & 36 & 21\\42 & 81 & 54\\41 & 62 & 61\end{matrix}\right]$

$\displaystyle A^{-1} =\left[\begin{matrix}- \frac{38}{33} & \frac{20}{33} & - \frac{1}{11}\\\frac{10}{33} & - \frac{7}{33} & \frac{2}{11}\\\frac{17}{33} & - \frac{2}{33} & - \frac{1}{11}\end{matrix}\right]$

$\displaystyle inv(A) =\left[\begin{matrix}- \frac{38}{33} & \frac{20}{33} & - \frac{1}{11}\\\frac{10}{33} & - \frac{7}{33} & \frac{2}{11}\\\frac{17}{33} & - \frac{2}{33} & - \frac{1}{11}\end{matrix}\right]$

---

Manipulação de matrizes

display(A.col(1))
print()
display(A.row(0))
print()
display(A.row_insert(2,Matrix([[9,9,9]])))

$\displaystyle \left[\begin{matrix}2\\5\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}1 & 2 & 3\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\9 & 9 & 9\end{matrix}\right]$

---

Propriedades

A = Matrix([
    [1,9,3],
    [5,2,-1],
    [2,0,-3]
])

display(Math('A ='+latex(A)))

display(Math('A shape ='+latex(A.shape)))
display(Math('A rank ='+latex(A.rank())))
display(Math('A det ='+latex(A.det())))

$\displaystyle A =\left[\begin{matrix}1 & 9 & 3\\5 & 2 & -1\\2 & 0 & -3\end{matrix}\right]$

$\displaystyle A shape =\left( 3, \ 3\right)$

$\displaystyle A rank =3$

$\displaystyle A det =99$

---

Matriz reduzida

A = Matrix([
    [1,9,3,11],
    [5,2,-1,20],
    [2,0,-3,-2]
])

display(Math('A ='+latex(A)))


display(Math('A ='+latex(A.rref())))

$\displaystyle A =\left[\begin{matrix}1 & 9 & 3 & 11\\5 & 2 & -1 & 20\\2 & 0 & -3 & -2\end{matrix}\right]$

$\displaystyle A =\left( \left[\begin{matrix}1 & 0 & 0 & \frac{56}{11}\\0 & 1 & 0 & - \frac{23}{33}\\0 & 0 & 1 & \frac{134}{33}\end{matrix}\right], \ \left( 0, \ 1, \ 2\right)\right)$

---

Matrizes elementares

display(eye(3))
display(zeros(3,4))
display(ones(2,4))

$\displaystyle \left[\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}1 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{matrix}\right]$

---

Autovetores e autovalores

A = Matrix([
    [ 3, 1,-4, 0],
    [-1,-3, 0, 0],
    [ 3, 1,-5,-1],
    [-3,-1, 5, 1]
])

print('Autovalores:',A.eigenvals(),'\n')

#(Autovalor, multiplicidade, autovetor)
print('Autovalores e autovetores:')
display(A.eigenvects())

Autovalores: {-2: 2, 0: 2} 

Autovalores e autovetores:

[(-2,
  2,
  [Matrix([
   [-1],
   [ 1],
   [-1],
   [ 1]])]),
 (0,
  2,
  [Matrix([
   [-3/2],
   [ 1/2],
   [  -1],
   [   1]])])]

---

Diagonalização

A = Matrix([
    [3, -2,  4, -2],
    [1,  3, -3, -2],
    [5, -2,  2, -2],
    [5, -2, -3,  3]])

# A = P * D *(P^-1)
display(Math('A ='+latex(A)))

P,D = A.diagonalize()
print()
display(Math('A = P * D * P^{-1}'))
print()
display(Math('P ='+latex(P)))
display(Math('D ='+latex(D)))

$\displaystyle A =\left[\begin{matrix}3 & -2 & 4 & -2\\1 & 3 & -3 & -2\\5 & -2 & 2 & -2\\5 & -2 & -3 & 3\end{matrix}\right]$

$\displaystyle A = P * D * P^{-1}$

$\displaystyle P =\left[\begin{matrix}0 & 1 & 0 & 1\\1 & 2 & -1 & -1\\1 & 1 & 0 & 1\\1 & 1 & 1 & 1\end{matrix}\right]$

$\displaystyle D =\left[\begin{matrix}-2 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 5 & 0\\0 & 0 & 0 & 7\end{matrix}\right]$

---

Sistema Linear

A = Matrix([
    [1,5,8,1],
    [2,-1,0,0],
    [1,1,1,5],
    [5,5,0,2]
])

b = Matrix([1,2,2,1])

display(Math('A ='+latex(A)))
display(Math('b ='+latex(b)))

print()

display(Math('A*x = b'))

print()

display(Math('x ='+latex(linsolve([A,b]))))

$\displaystyle A =\left[\begin{matrix}1 & 5 & 8 & 1\\2 & -1 & 0 & 0\\1 & 1 & 1 & 5\\5 & 5 & 0 & 2\end{matrix}\right]$

$\displaystyle b =\left[\begin{matrix}1\\2\\2\\1\end{matrix}\right]$

$\displaystyle A*x = b$

$\displaystyle x =\left\{\left( \frac{9}{13}, \ - \frac{8}{13}, \ \frac{5}{13}, \ \frac{4}{13}\right)\right\}$

---

Sistema Linear subdeterminado

A = Matrix([
    [1,5,8,1],
    [2,-1,0,0],
    [5,5,0,2]
])

b = Matrix([1,2,2,1])

display(Math('A ='+latex(A)))
display(Math('b ='+latex(b)))

print()

display(Math('A*x = b'))

print()

soluc = linsolve([A,b],[x,y,z,t])
display(Math('x ='+latex(soluc)))

print()

display('t = 1',soluc.subs('t',1))

$\displaystyle A =\left[\begin{matrix}1 & 5 & 8 & 1\\2 & -1 & 0 & 0\\5 & 5 & 0 & 2\end{matrix}\right]$

$\displaystyle b =\left[\begin{matrix}1\\2\\2\\1\end{matrix}\right]$

$\displaystyle A*x = b$

$\displaystyle x =\left\{\left( \frac{4}{5} - \frac{2 t}{15}, \ - \frac{4 t}{15} - \frac{2}{5}, \ \frac{7 t}{120} + \frac{11}{40}, \ t\right)\right\}$

't = 1'

$\displaystyle \left\{\left( \frac{2}{3}, \ - \frac{2}{3}, \ \frac{1}{3}, \ 1\right)\right\}$

---

Sem solução

A = Matrix([
    [2,2],
    [1,1],
])

b = Matrix([2,2])

display(Math('A ='+latex(A)))
display(Math('b ='+latex(b)))

print()

display(Math('A * x = b'))

print()

display(Math('x =' + latex(linsolve([A,b]))))

$\displaystyle A =\left[\begin{matrix}2 & 2\\1 & 1\end{matrix}\right]$

$\displaystyle b =\left[\begin{matrix}2\\2\end{matrix}\right]$

$\displaystyle A * x = b$

$\displaystyle x =\emptyset$