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Linear Algebra
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m = MatrixPower[a, 5]+ 2 MatrixPower[a, 4] MatrixPower[a, 3]+ MatrixPower[a, 2] 3 a + 2 IdentityMatrix[5] ; MatrixForm[m, TableAlignments Right] 496 948 189 1776 1695 726 862 288 714 66 117 399 103 648 1233 9 174 324 315 1216 1875 2 1419 1068 267 702 1069 a 12.16 It can be shown that the complex number a + bi and the matrix b have the same algebraic b a
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properties. Compute (2 + 3i ) 5 using matrices and verify using complex arithmetic that this value is correct.
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2 3 a= ; 3 2 MatrixPower[a, 5]; 122 597 597 122 (2 + 3I)5 122 597
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This represents the number 122 597 i.
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12.17 Compute the determinants 1 x1 x12 . . x1n 1 1 x2 2 x2 . . n x 2 1 1 x3 2 x3 . . n x3 1 . . . . . . . 1 . xn 2 . xn . . . . n . x n 1
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for n = 2, 3, 4, and 5. Can you determine a pattern These are known as Vandermonde determinants.
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m[n_] Table[x[i]^j, {j, 0, n 1}, {i, 1, n}]; m[2] //MatrixForm 1 1 x[1] x[2] m[3] //MatrixForm 1 1 1 x[1] x[2] x[3] x[1]2 x[2]2 x[3]2 Det[m[2]] // Factor x[1]+ x[2] Det[m[3]] // Factor (x[1] x[2])(x[1] x[3])(x[2] x[3]) Det[m[4]] // Factor (x[1] x[2])(x[1] x[3])(x[2] x[3])(x[1] x[4])(x[2] x[4])(x[3] x[4])
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Linear Algebra
Det[m[5]] // Factor (x[1] x[2])(x[1] x[3])(x[2] x[3])(x[1] x[4]) (x[2] x[4])(x[3] x[4])(x[1] x[5])(x[2] x[5]) (x[3] x[5])(x[4] x[5]) In general Det[ m[n] = ( x[i] x[j] . ] )
12.18 A theorem of linear algebra says that the determinant of a matrix is the sum of the products of each entry of any row or column by its corresponding cofactor. (The cofactor, Ci j , of ai j is ( 1)i + j Mi j where Mi j is the corresponding minor.) Use this to compute the determinant of a randomly generated 5 5 matrix and verify its value.
SOLUTION
n = 5; a = TableRandomInteger[9], {i, 1, n}, {j, 1, n}]; a // MatrixForm 4 5 1 7 0 6 3 6 9 9 5 0 9 7 4 3 5 7 1 5 3 6 7 2 6
matrixofminors = Minors[a]; MatrixForm[matrixofminors, TableAlignments Right] 159 174 171 1350 1584 270 669 549 342 4 96 561 339 8 3231 3333 438 283 216 48 293 78 143 168 497 246
140
signs = Table[( 1)^(i + j), {i, 1, n}, {j, 1, n}]; cofactors = matrixofminors * signs; MatrixForm[cofactors, TableAlignments Right] 171 159 174 283 216 1350 1584 270 140 48 669 549 342 293 78 561 339 96 143 168 7 3231 3333 438 497 246 i=3 (* we expand using the third row *)
determinant = a[[n i + 1,n j+ 1]]* cofactors[[i,j]] ]
2082 Det[a] 2082
The (i, j)th element of Minors[a] gives the determinant of the matrix obtained by deleting row n i + 1 and column n j + 1.
Linear Algebra
1 1 1 2 2 2 12.19 Let x = 3 . Compute x T x = [1 2 3 4 5] 3 and xx T = 3 [1 2 3 4 5] . 4 4 4 5 5 5
SOLUTION
xTx is the dot product of the vector x with itself. xxT, however, is a 5 5 matrix. x.x 55 Outer[Times, x, x] // MatrixForm 1 2 3 4 2 4 6 8 3 6 9 12 4 8 12 16 5 10 15 20 5 10 15 20 25
12.3 Matrix Manipulation
Mathematica offers a variety of matrix manipulation commands that are quite useful when working problems in linear algebra. Since matrices are actually lists, many of the commands are the same as described in 3.
Join[list1, list2] combines the two lists list1 and list2 into one list consisting of the elements from list1 and from list2. For matrices, this has the effect of placing the rows of list2 under the rows of list1. Join[list1, list2, n] joins the objects at level n in each list. If n = 2, this has the effect of placing the columns of list2 to the right of the columns of list1. ArrayFlatten[{{m11, m12,...}, {m21, m22,...},...}]creates a single flattened matrix from a matrix of matrices mi j. All the matrices in the same row must have the same first dimension, and all the matrices in the same column must have the same second dimension.
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