restart;with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

P:= Matrix([[2,4],[3,7]]);

NiM+SSJQRzYiLUknUlRBQkxFR0YlNiUiJ2soZSMtSSdNQVRSSVhHRiU2IzckNyQiIiMiIiU3JCIiJCIiKEknTWF0cml4RzYkSSpwcm90ZWN0ZWRHRjZJKF9zeXNsaWJHRiU=

v:= Matrix([[1],[1],[1]]);

NiM+SSJ2RzYiLUknUlRBQkxFR0YlNiUiKHcyWyItSSdNQVRSSVhHRiU2IzclNyMiIiJGLkYuSSdNYXRyaXhHNiRJKnByb3RlY3RlZEdGMkkoX3N5c2xpYkdGJQ==

M:= 10*Matrix(3,3,shape=identity);

NiM+SSJNRzYiLUknUlRBQkxFR0YlNiUiJz9JPy1JJ01BVFJJWEdGJTYjNyU3JSIjNSIiIUYwNyVGMEYvRjA3JUYwRjBGL0knTWF0cml4RzYkSSpwcm90ZWN0ZWRHRjVJKF9zeXNsaWJHRiU=

B:= Matrix([[3,3,3],[2,2,2,2],[1,1,1]],shape=band[1,1],scan=band[1,1]);

NiM+SSJCRzYiLUknUlRBQkxFR0YlNiUiJ086aS1JJ01BVFJJWEdGJTYjNyY3JiIiIyIiIiIiIUYxNyYiIiRGL0YwRjE3JkYxRjNGL0YwNyZGMUYxRjNGL0knTWF0cml4RzYkSSpwcm90ZWN0ZWRHRjhJKF9zeXNsaWJHRiU=

Mv:= M.v; # matrix vector multiplication

NiM+SSNNdkc2Ii1JJ1JUQUJMRUdGJTYlIihfb0UiLUknTUFUUklYR0YlNiM3JTcjIiM1Ri5GLkknTWF0cml4RzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRiU=

A:= Matrix([[1,2,3],[3,2,1],[1,3,2]]);

NiM+SSJBRzYiLUknUlRBQkxFR0YlNiUiJ3dFbC1JJ01BVFJJWEdGJTYjNyU3JSIiIiIiIyIiJDclRjFGMEYvNyVGL0YxRjBJJ01hdHJpeEc2JEkqcHJvdGVjdGVkR0Y2SShfc3lzbGliR0Yl

MA:= M.A; # matrix-matrix multiplication

NiM+SSNNQUc2Ii1JJ1JUQUJMRUdGJTYlIidLJCpmLUknTUFUUklYR0YlNiM3JTclIiM1IiM/IiNJNyVGMUYwRi83JUYvRjFGMEknTWF0cml4RzYkSSpwcm90ZWN0ZWRHRjZJKF9zeXNsaWJHRiU=

Ainv:= Matrix(inverse(A)); # inverse of matrix A

NiM+SSVBaW52RzYiLUknUlRBQkxFR0YlNiUiKD87RCQtSSdNQVRSSVhHRiU2IzclNyUjIiIiIiM3IyIiJkYxIyEiIiIiJDclIyEiJkYxI0Y1RjEjIiIjRjY3JSMiIihGMUY6RjRJJ01hdHJpeEc2JEkqcHJvdGVjdGVkR0ZCSShfc3lzbGliR0Yl

lambda:= eigenvals(A); # eigenvalues of A

NiM+SSdsYW1iZGFHNiI2JSIiJywmIyEiIiIiIyIiIiomXiMjRixGK0YsIiIoRi9GLCwmRilGLComXiNGKUYsRjBGL0Ys

v:= eigenvectors(A);

NiM+SSJ2RzYiNiU3JSIiJyIiIjwjLUkndmVjdG9yRzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiU2IzclRilGKUYpNyUsJiMhIiIiIiNGKSomXiMjRilGNkYpIiIoRjlGKUYpPCMtRiw2IzclLCYjRikiI1dGKSomXiMjISM8RkFGKUY6RjlGKSwmIyEjUEZBRikqJl4jIyIjOEZBRilGOkY5RilGKTclLCZGNEYpKiZeI0Y0RilGOkY5RilGKTwjLUYsNiM3JSwmRkBGKSomXiMjIiM8RkFGKUY6RjlGKSwmRkdGKSomXiMjISM4RkFGKUY6RjlGKUYp

# eigenvectors give the eigenvalues and eigenvectors of A. The first entry[6,1,{[1,1,1]}] tells you the eigenvalue 6 has multiplicity 1 with corresponding eigenvector [1,1,1] TTdSMApJM1JUQUJMRV9TQVZFLzI1ODc2NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIjIiIkIiIlIiIoRiYKTTdSMApJNFJUQUJMRV9TQVZFLzE0ODA3NzZYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMkIiQiIiIiIkYnRidGJgo=TTdSMApJM1JUQUJMRV9TQVZFLzIwMzAyMFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyoiJCIkIiM1IiIhRihGKEYnRihGKEYoRidGJgo=TTdSMApJM1JUQUJMRV9TQVZFLzYyMTUzNlguJSlhbnl0aGluZ0c2IyYlJWJhbmRHNiQiIiJGKTYiW2dsISIkISEhIy0iJSIlIiIiIiEiIiMiIiRGCilGLEYtRilGLEYtRilGLEYrRioKTTdSMApJNFJUQUJMRV9TQVZFLzEyNjY4NTJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMkIiQiIiIjNUYnRidGJgo=TTdSMApJM1JUQUJMRV9TQVZFLzY1MjY3NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyoiJCIkIiIiIiIkRiciIiNGKUYoRihGJ0YpRiYKTTdSMApJM1JUQUJMRV9TQVZFLzU5OTMzMlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyoiJCIkIiM1IiNJRiciIz9GKUYoRihGJ0YpRiYKTTdSMApJNFJUQUJMRV9TQVZFLzMyNTE2MjBYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCMiIiIiIzcjISImRikjIiIoRikjIiImRikjCiEiIkYpRjAjRjEiIiQjIiIjRjNGMkYmCg==