There are three states under consideration: Texas, Oklahome, and New Mexico. We consider a stable population size, and just consider migrations between the states. Each year, 0.9 of the Texas population stays in Texas, 0.08 of the Texas population moves to Oklahoma, and 0.02 of the Texas populationmoves to New Mexico. For Oklahoma, 0.2 will move to Texas, 0.7 will stay in Oklahoma, and 0.1 will move to New Mexico. For New Mexico: 0.1 will move toTexas, 0.1 will move to Oklahoma, and 0.8 will stay in New Mexico.NiNJJlRleGFzRzYiNiNJKU9rbGFob21hRzYiNiNJKk5ld01leGljb0c2Ig==NiNJJlRleGFzRzYiNiNJKU9rbGFob21hRzYiNiNJKk5ld01leGljb0c2Ig==NiNJJUZyb21HNiI=NiNJI1RvRzYiNiMkIiIqISIiNiMkIiIpISIjNiMkIiIjISIjNiMkIiIjISIiNiMkIiIoISIiNiMkIiIiISIiNiMkIiIiISIiNiMkIiIiISIiNiMkIiIpISIiLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We can express the migration data as a matrix:QyQtSSV3aXRoRzYiNiNJLkxpbmVhckFsZ2VicmFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQ+SSJBRzYiLUknTWF0cml4R0YlNiM3JTclJCIiKiEiIiQiIiNGLSQiIiJGLTclJCIiKSEiIyQiIihGLUYwNyUkRi9GNUYwJEY0Ri1GMQ==print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=If x is the population vector for a given year, then A x gives the population vector for the following year. Thus, A^n x gives the population vector after n years. Let's suppose the initial populations of the states are: Texas 30, Oklahoma 10, New Mexico 8 (in millions). After one year:QyQ+SSJ4RzYiLUknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM3JSIjSSIjNSIiKSIiIg==print(); # input placeholderQyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiRJIkFHRilJInhHRikiIiI=print(); # input placeholder After 2 years:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQqJEkiQUdGKSIiI0kieEdGKSIiIg==print(); # input placeholderAfter 10 years:QyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQqJEkiQUdGKSIjNUkieEdGKSIiIg==print(); # input placeholderAfter 100 years:QyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQqJEkiQUdGKSIkKyJJInhHRikiIiI=print(); # input placeholderAfter 500 years:QyQ+SSJCRzYiLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHNiRGKUkoX3N5c2xpYkdGJTYkKiRJIkFHRiUiJCsmSSJ4R0YlIiIiprint(); # input placeholderWe see the populations tens to a stable distribution. Note that the total population does not change. JSFHJSFHJSFHWe compute the eigenvalues and eigenvectotrs for the matrix A.JSFHJSFHJSFHQyQtSSxFaWdlbnZhbHVlc0c2IjYjSSJBR0YlIiIiprint(); # input placeholderJSFHQyQ+SSJFRzYiLUktRWlnZW52ZWN0b3JzR0YlNiNJIkFHRiUiIiI=print(); # input placeholderJSFHJSFHWe normalize the population vector so its entries sum to 1. JSFHJSFHQyQqJkkiQkc2IiIiIiwoLUYkNiRGJkYmRiYtRiQ2JCIiI0YmRiYtRiQ2JCIiJEYmRiYhIiJGJg==print(); # input placeholderJSFHWe isolate the eigenvector for the eigenvalue lambda=1. We normalize it so its entries sum to 1.JSFHJSFHJSFHQyQ+SSJGRzYiLUknQ29sdW1uR0YlNiQmSSJFR0YlNiMiIiMiIiJGLQ==print(); # input placeholderQyQqJkkiRkc2IiIiIiwoLUYkNiRGJkYmRiYtRiQ2JCIiI0YmRiYtRiQ2JCIiJEYmRiYhIiJGJg==print(); # input placeholderJSFHJSFHNote that all the entries of this eigenvector are positive. Note that the other eigenvalues are less than 1 in absolute value. We see the long term stable population is (up to a scalar constant) the eigenvector corresponding to the eigenvalue lambda=1.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Next we compute some powers of the matrix A. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyRJIkFHNiIiIiI=print(); # input placeholderJSFHJSFHQyQqJEkiQUc2IiIiJiIiIg==print(); # input placeholderQyQqJEkiQUc2IiIjNSIiIg==print(); # input placeholderJSFHQyQqJEkiQUc2IiIjXSIiIg==print(); # input placeholderJSFHJSFHJSFHQyQ+SSJHRzYiKiRJIkFHRiUiJCsiIiIiprint(); # input placeholderQyQtSSdDb2x1bW5HNiI2JEkiR0dGJSIiIkYoprint(); # input placeholderWe see that A^n converges to a matrix all of whose columns in the (stochastic) eigenvector corresponding to lambda=1.The facts seen in this example are true for any positive stochastic matrix (i.e., where the columns sum to 1). TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzM5Mjg5Njc4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQkIiIqISIiJCIiKSEiIyQiIiNGKyRGLUYoJCIiKEYoJCIiIkYoRjFGMSRGKkYoRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzM5Mjg5OTE4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiNJIiM1IiIpRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzM5MjkwMjc4WColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkNDAzRENDQ0NDQ0NDQ0NDRDQwMjQ2NjY2NjY2NjY2Njc0MDIwMDAwMDAwMDAwMDAwRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzM5Mjg3MTU4WColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkNDAzREE4RjVDMjhGNUMyQTQwMjRBNUUzNTNGN0NFRDg0MDIwMDgzMTI2RTk3OEQ2RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzU4NDYyWColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkNDAzRDUwOTY0MEM3MTJEMDQwMjUwRTg2NzM0QjM4Mjc0MDIwNTA0RDBCMjZBMjQyRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzU5MzAyWColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkNDAzRDQ0QUVENDRCMDk4QzQwMjUxMkJCNTEyQkFGMzI0MDIwNjNFNzA2M0UzRTBBRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzYwMTQyWColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkNDAzRDQ0QUVENDRBRURDNTQwMjUxMkJCNTEyQkI1NkY0MDIwNjNFNzA2M0U3MEFDRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzYwODYyWColKWFueXRoaW5nRzYiRiVbZ2woIyUhISEiJCIkM0ZGMDAwMDAwMDAwMDAwMTAwMDAwMDAwMDAwMDAwMDAzRkU5NDMxRDYwOTMzMTYzMDAwMDAwMDAwMDAwMDAwMDNGRTM4OUFGNkMzOTlCNjgwMDAwMDAwMDAwMDAwMDAwRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzU0ODQ2WColKWFueXRoaW5nRzYiRiVbZ2woIyUhISEiJCIkM0ZGMDAwMDAwMDAwMDAwMTAwMDAwMDAwMDAwMDAwMDAzRkU5NDMxRDYwOTMzMTYzMDAwMDAwMDAwMDAwMDAwMDNGRTM4OUFGNkMzOTlCNjgwMDAwMDAwMDAwMDAwMDAwRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzU2NTI2WCwlKWFueXRoaW5nRzYiRiVbZ2woIiUhISEjKiIkIiRCRkVEMUQ2RjY1M0QxMTY3MDAwMDAwMDAwMDAwMDAwMEJGRDRGNjc5MkEyQkY4MDkwMDAwMDAwMDAwMDAwMDAwQkZEMDRERUM3NjIyMzJCNjAwMDAwMDAwMDAwMDAwMDBCRkU3QzcyM0JBMkFFOTdFMDAwMDAwMDAwMDAwMDAwMDNGQjQxNTE3NjQ3M0JCNUMwMDAwMDAwMDAwMDAwMDAwM0ZFNTQ0ODBDRDlDNzIxRjAwMDAwMDAwMDAwMDAwMDAzRkRCOTFENkE2OTVCNjVEMDAwMDAwMDAwMDAwMDAwMEJGRUExRDQ3RDJGMUU1QjAwMDAwMDAwMDAwMDAwMDAwM0ZEOEE4QjhGRjRFMTRGNjAwMDAwMDAwMDAwMDAwMDBGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzQ5NTUwWColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkM0ZFMzgzMUYzODMxRjM4MzNGQ0MxOEY5QzE4RjlDMTkzRkM1REE4OTVEQTg5NURCRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzU2NzY2WColKWFueXRoaW5nRzYiRiVbZ2woIyUhISEiJCIkQkZFRDFENkY2NTNEMTE2NzAwMDAwMDAwMDAwMDAwMDBCRkQ0RjY3OTJBMkJGODA5MDAwMDAwMDAwMDAwMDAwMEJGRDA0REVDNzYyMjMyQjYwMDAwMDAwMDAwMDAwMDAwRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzQ5NjcwWColKWFueXRoaW5nRzYiRiVbZ2woIyUhISEiJCIkM0ZFMzgzMUYzODMxRjM4MzAwMDAwMDAwMDAwMDAwMDAzRkNDMThGOUMxOEY5QzE0MDAwMDAwMDAwMDAwMDAwMDNGQzVEQTg5NURBODk1REMwMDAwMDAwMDAwMDAwMDAwRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzQ5NzkwWCwlKWFueXRoaW5nRzYiRiVbZ2wnIiUhISEjKiIkIiQzRkU2QTI0MURFQzdBMTE1M0ZDOEQ3QzE0Nzg4N0ZDMTNGQjkzRTZFN0FCMUY3RTEzRkUwNUVDREVCQjBCREEzM0ZEMjk5Q0NDNTQwMTA5MDNGQzk1MTJFQzZCQ0U4NTUzRkQ4Q0U2MTIxODMwNzg5M0ZDQzA0M0YyOUUxN0FEQTNGRDkyRjdGNDk4QzNCMEZGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzUwMTUwWCwlKWFueXRoaW5nRzYiRiVbZ2wnIiUhISEjKiIkIiQzRkU0NjhDOUY2MjNFODVEM0ZDQjhERUQzMkY2QTBGODNGQzJDRUVBRjQ3OUJEQTAzRkUyQ0EzNTc3OUI0RkQ2M0ZDRDBGRkQxRDAwMDNENzNGQzdDNzJEMDQ5MkJDRDMzRkUxM0NBMEQyOTM4MDE4M0ZDQ0NCRkRCNkZDRTBDNTNGQ0U0MTdFRkVCNTFFRUNGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzUwODcwWCwlKWFueXRoaW5nRzYiRiVbZ2wnIiUhISEjKiIkIiQzRkUzODMyM0IyMTVFMzI2M0ZDQzE4RjdEREE3NzNDNDNGQzVEQTc5NUEwMEZGREUzRkUzODMxQkZFM0UzQTZDM0ZDQzE4RkIxRTYyMEQ3QTNGQzVEQTk0RThBNTA5MDkzRkUzODMxMzYxQUFGNzkxM0ZDQzE4RkVDMTUwNzYxMTNGQzVEQUIzQjgwM0FCRUZGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzU3MzY2WCwlKWFueXRoaW5nRzYiRiVbZ2wnIiUhISEjKiIkIiQzRkUzODMxRjM4MzQxQjM3M0ZDQzE4RjlDMThFQjMzRTNGQzVEQTg5NURBMEUwNTQzRkUzODMxRjM4MzA2NUY2M0ZDQzE4RjlDMTkwNDQxOTNGQzVEQTg5NURBRTI0N0EzRkUzODMxRjM4MkM0MEFCM0ZDQzE4RjlDMTkyMDQ0OTNGQzVEQTg5NURCQ0Y5ODBGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY2MzUwOTkwWColKWFueXRoaW5nRzYiRiVbZ2wnIyUhISEiJCIkM0ZFMzgzMUYzODM0MUIzNzNGQ0MxOEY5QzE4RUIzM0UzRkM1REE4OTVEQTBFMDU0RiU=