added some exercises

ex_41.py

@@ -0,0 +1,20 @@
+import numpy as np
+
+A = np.array([ [1, 5, 1]
+		, [5, 0, -3]
+		, [1, -3, 7]])
+
+print(np.linalg.eig(A))
+
+O = np.linalg.eig(A)[1]
+D = np.zeros((3, 3))
+for i, v in enumerate(np.linalg.eig(A)[0]):
+	D[i][i] = v
+
+I = np.zeros((3, 3))
+for i in range(3):
+	I[i][i] = 1
+
+print("(2)", np.allclose(np.dot(O.T, O), I))
+print("(3)", np.allclose(A, np.dot(np.dot(O, D), O.T)))
+print("(4)", np.allclose(np.dot(A, A), np.dot(np.dot(O, np.dot(D, D)), O.T)))

ex_42.py

@@ -0,0 +1,25 @@
+import numpy as np
+
+
+A = np.array(
+	[ [0, 1, 1, 0, 0, 0]
+	, [1, 0, 1, 0, 1, 0]
+	, [1, 1, 0, 1, 0, 0]
+	, [0, 0, 1, 0, 1, 0]
+	, [0, 0, 0, 1, 0, 1]
+	, [0, 0, 0, 0, 1, 0]
+	])
+
+eigvalues, eigvectors = np.linalg.eig(A)
+
+l_max_i = eigvalues.argmax()
+l_max = eigvalues[l_max_i]
+v_max = eigvectors[l_max_i]
+
+def some_norm(M):
+	return np.abs(M).max()
+
+for k in range(1, 21):
+	print("some kind of error for k =", k, ":", some_norm(A**k - l_max**k * np.outer(v_max, v_max.T)))
+
+

ex_43.py

@@ -0,0 +1,15 @@
+import numpy as np
+
+P = np.array(
+	[ [1/6, 1/6, 1/6, 1/6, 1/6, 0]
+	, [1/6, 1/6, 1/6, 1/6, 1/6, 0]
+	, [1/6, 1/6, 1/6, 1/6, 1/6, 0]
+	, [1/6, 1/6, 1/6, 1/6, 1/6, 0]
+	, [1/6, 1/6, 1/6, 1/6, 1/6, 1/10]
+	, [1/6, 1/6, 1/6, 1/6, 1/6, 9/10]])
+print(P.shape)
+
+v = np.array([0, 1, 0, 0, 0, 0])
+w = np.power(np.dot(P**4, v), 1/4)
+print(w)
+print(w[-1])