Matrices eigenvalues

From SymPy

<source lang="python"> In [1]: M = Matrix(4, 4, lambda i,j: i*j+1)

In [2]: M.eigenvals() Out[2]: ⎧ ⎽⎽⎽⎽ ⎽⎽⎽⎽ ⎫ ⎨9 - ╲╱ 61 : 1, 0: 2, 9 + ╲╱ 61 : 1⎬ ⎩ ⎭

In [3]: M = Matrix(4, 4, lambda i,j: i*j+2)

In [4]: M.eigenvals() Out[4]: {2: 1, 0: 2, 20: 1}

In [5]: M = Matrix(4, 4, lambda i,j: i*j+3)

In [6]: M.eigenvals() Out[6]: ⎧ ⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽ ⎫ ⎨13 - ╲╱ 109 : 1, 13 + ╲╱ 109 : 1, 0: 2⎬ ⎩ ⎭

In [7]: M Out[7]: ⎡ 3 3 3 3⎤ ⎢ 3 4 5 6⎥ ⎢ 3 5 7 9⎥ ⎣ 3 6 9 12⎦

In [8]: M = Matrix(4, 4, lambda i,j: i*j+x)

In [9]: M Out[9]: ⎡ x x x x⎤ ⎢ x 1 + x 2 + x 3 + x⎥ ⎢ x 2 + x 4 + x 6 + x⎥ ⎣ x 3 + x 6 + x 9 + x⎦

In [10]: M.eigenvals() Out[10]: ⎧ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎫ ⎪ ╱ 2 ╱ 2 ⎪ ⎨ ╲╱ 196 + 32*x + 16*x ╲╱ 196 + 32*x + 16*x ⎬ ⎪7 + 2*x + ───────────────────────: 1, 0: 2, 7 + 2*x - ───────────────────────: 1⎪ ⎩ 2 2 ⎭

</source> Such things are really tedious to do by hand, but in SymPy one can do them just fine.