Stochastic Control & Optimzation - Homework 1

1. Problem 1: Use definitions to prove

(A−1)T = (AT )−1, where A is an invertible square matrix and AT means the transpose of matrix A.

2. Calculate the lending strategy for the described problem using matrix inversion:

         [,1]
[1,] 76.38889
[2,] 62.50000
[3,] 31.94444
[4,] 79.16667

3. Formulate this as a Linear Programming problem with appropriate decision variables, constraints, and an objective. Do not solve the program.

4. a. Generating Lehmer matrix

            [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]      [,8]
 [1,] 1.00000000 0.5000000 0.3333333 0.2500000 0.2000000 0.1666667 0.1428571 0.1250000
 [2,] 0.50000000 1.0000000 0.6666667 0.5000000 0.4000000 0.3333333 0.2857143 0.2500000
 [3,] 0.33333333 0.6666667 1.0000000 0.7500000 0.6000000 0.5000000 0.4285714 0.3750000
 [4,] 0.25000000 0.5000000 0.7500000 1.0000000 0.8000000 0.6666667 0.5714286 0.5000000
 [5,] 0.20000000 0.4000000 0.6000000 0.8000000 1.0000000 0.8333333 0.7142857 0.6250000
 [6,] 0.16666667 0.3333333 0.5000000 0.6666667 0.8333333 1.0000000 0.8571429 0.7500000
 [7,] 0.14285714 0.2857143 0.4285714 0.5714286 0.7142857 0.8571429 1.0000000 0.8750000
 [8,] 0.12500000 0.2500000 0.3750000 0.5000000 0.6250000 0.7500000 0.8750000 1.0000000
 [9,] 0.11111111 0.2222222 0.3333333 0.4444444 0.5555556 0.6666667 0.7777778 0.8888889
[10,] 0.10000000 0.2000000 0.3000000 0.4000000 0.5000000 0.6000000 0.7000000 0.8000000
[11,] 0.09090909 0.1818182 0.2727273 0.3636364 0.4545455 0.5454545 0.6363636 0.7272727
[12,] 0.08333333 0.1666667 0.2500000 0.3333333 0.4166667 0.5000000 0.5833333 0.6666667
[13,] 0.07692308 0.1538462 0.2307692 0.3076923 0.3846154 0.4615385 0.5384615 0.6153846
[14,] 0.07142857 0.1428571 0.2142857 0.2857143 0.3571429 0.4285714 0.5000000 0.5714286
[15,] 0.06666667 0.1333333 0.2000000 0.2666667 0.3333333 0.4000000 0.4666667 0.5333333
[16,] 0.06250000 0.1250000 0.1875000 0.2500000 0.3125000 0.3750000 0.4375000 0.5000000
[17,] 0.05882353 0.1176471 0.1764706 0.2352941 0.2941176 0.3529412 0.4117647 0.4705882
[18,] 0.05555556 0.1111111 0.1666667 0.2222222 0.2777778 0.3333333 0.3888889 0.4444444
[19,] 0.05263158 0.1052632 0.1578947 0.2105263 0.2631579 0.3157895 0.3684211 0.4210526
[20,] 0.05000000 0.1000000 0.1500000 0.2000000 0.2500000 0.3000000 0.3500000 0.4000000
           [,9]     [,10]      [,11]      [,12]      [,13]      [,14]      [,15]
 [1,] 0.1111111 0.1000000 0.09090909 0.08333333 0.07692308 0.07142857 0.06666667
 [2,] 0.2222222 0.2000000 0.18181818 0.16666667 0.15384615 0.14285714 0.13333333
 [3,] 0.3333333 0.3000000 0.27272727 0.25000000 0.23076923 0.21428571 0.20000000
 [4,] 0.4444444 0.4000000 0.36363636 0.33333333 0.30769231 0.28571429 0.26666667
 [5,] 0.5555556 0.5000000 0.45454545 0.41666667 0.38461538 0.35714286 0.33333333
 [6,] 0.6666667 0.6000000 0.54545455 0.50000000 0.46153846 0.42857143 0.40000000
 [7,] 0.7777778 0.7000000 0.63636364 0.58333333 0.53846154 0.50000000 0.46666667
 [8,] 0.8888889 0.8000000 0.72727273 0.66666667 0.61538462 0.57142857 0.53333333
 [9,] 1.0000000 0.9000000 0.81818182 0.75000000 0.69230769 0.64285714 0.60000000
[10,] 0.9000000 1.0000000 0.90909091 0.83333333 0.76923077 0.71428571 0.66666667
[11,] 0.8181818 0.9090909 1.00000000 0.91666667 0.84615385 0.78571429 0.73333333
[12,] 0.7500000 0.8333333 0.91666667 1.00000000 0.92307692 0.85714286 0.80000000
[13,] 0.6923077 0.7692308 0.84615385 0.92307692 1.00000000 0.92857143 0.86666667
[14,] 0.6428571 0.7142857 0.78571429 0.85714286 0.92857143 1.00000000 0.93333333
[15,] 0.6000000 0.6666667 0.73333333 0.80000000 0.86666667 0.93333333 1.00000000
[16,] 0.5625000 0.6250000 0.68750000 0.75000000 0.81250000 0.87500000 0.93750000
[17,] 0.5294118 0.5882353 0.64705882 0.70588235 0.76470588 0.82352941 0.88235294
[18,] 0.5000000 0.5555556 0.61111111 0.66666667 0.72222222 0.77777778 0.83333333
[19,] 0.4736842 0.5263158 0.57894737 0.63157895 0.68421053 0.73684211 0.78947368
[20,] 0.4500000 0.5000000 0.55000000 0.60000000 0.65000000 0.70000000 0.75000000
          [,16]      [,17]      [,18]      [,19] [,20]
 [1,] 0.0625000 0.05882353 0.05555556 0.05263158  0.05
 [2,] 0.1250000 0.11764706 0.11111111 0.10526316  0.10
 [3,] 0.1875000 0.17647059 0.16666667 0.15789474  0.15
 [4,] 0.2500000 0.23529412 0.22222222 0.21052632  0.20
 [5,] 0.3125000 0.29411765 0.27777778 0.26315789  0.25
 [6,] 0.3750000 0.35294118 0.33333333 0.31578947  0.30
 [7,] 0.4375000 0.41176471 0.38888889 0.36842105  0.35
 [8,] 0.5000000 0.47058824 0.44444444 0.42105263  0.40
 [9,] 0.5625000 0.52941176 0.50000000 0.47368421  0.45
[10,] 0.6250000 0.58823529 0.55555556 0.52631579  0.50
[11,] 0.6875000 0.64705882 0.61111111 0.57894737  0.55
[12,] 0.7500000 0.70588235 0.66666667 0.63157895  0.60
[13,] 0.8125000 0.76470588 0.72222222 0.68421053  0.65
[14,] 0.8750000 0.82352941 0.77777778 0.73684211  0.70
[15,] 0.9375000 0.88235294 0.83333333 0.78947368  0.75
[16,] 1.0000000 0.94117647 0.88888889 0.84210526  0.80
[17,] 0.9411765 1.00000000 0.94444444 0.89473684  0.85
[18,] 0.8888889 0.94444444 1.00000000 0.94736842  0.90
[19,] 0.8421053 0.89473684 0.94736842 1.00000000  0.95
[20,] 0.8000000 0.85000000 0.90000000 0.95000000  1.00

4.b. Test whether A is symmetric or not. Namely, is A equal to AT?

Yes, the two matrices are equal as the maximum difference between any two elements in the original matrix and the transposed matrix is zero

[1] 0

4.c. Calculate C = A−1 in R. Test whether the inverse is correct. That is, calculate C ×A in R and see whether the product is an identity matrix.

[1] TRUE

4.d. Assign [1,2,3,4,5,6,7,8,9,10,10,9,8,7,6,5,4,3,2,1] to d in R.

      [,1]
 [1,]    1
 [2,]    2
 [3,]    3
 [4,]    4
 [5,]    5
 [6,]    6
 [7,]    7
 [8,]    8
 [9,]    9
[10,]   10
[11,]   10
[12,]    9
[13,]    8
[14,]    7
[15,]    6
[16,]    5
[17,]    4
[18,]    3
[19,]    2
[20,]    1

4.e. Solve for x in the equation Ax = Cd.

               [,1]
 [1,] -5.304478e-15
 [2,]  1.248443e-14
 [3,]  1.737561e-14
 [4,] -7.210962e-14
 [5,]  9.849542e-15
 [6,] -1.489992e-14
 [7,]  1.127494e-13
 [8,] -3.373619e-15
 [9,] -2.481203e+01
[10,]  2.006424e+01
[11,]  3.581375e+01
[12,] -3.006263e+01
[13,] -3.736996e-04
[14,] -2.772044e-04
[15,] -2.099688e-04
[16,] -1.619541e-04
[17,] -1.269228e-04
[18,] -1.008779e-04
[19,]  9.505933e+01
[20,] -1.000629e+02