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EXAMPLE 7
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ConstantArray[0, {3, 5}] //MatrixForm 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 HilbertMatrix[5] //MatrixForm 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7 1 4 1 5 1 6 1 7 1 8 1 5 1 6 1 7 1 8 1 9
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Linear Algebra
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HankelMatrix[{a, b, c, d, e}] //MatrixForm a b c d e b c d e 0 c d e 0 0 d e 0 0 0 e 0 0 0 0
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12.1 Construct a ten-dimensional vector of powers of 2.
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powersof2 = Table[2k, {k, 1, 10}]
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4, 8, 16, 32, 64, 128, 256, 512, 1024}
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powersof2 // MatrixForm 2 4 8 16 32 64 128 256 512 1024
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12.2 Construct a 5 5 matrix of random digits.
SOLUTION
Table[RandomInteger[9], {i, 5}, {j, 5}] // MatrixForm 5 2 1 7 9 7 8 0 8 2 9 6 8 8 2 9 9 2 8 3 4 7 2 1 4
What happens if //MatrixForm is included within the definition of a matrix
SOLUTION
m1 = {{1, 1}, {1, 2}} //MatrixForm m2 = {{2, 3}, {4, 5}} //MatrixForm 1 1 1 2 2 3 4 5 m1 + m2 1 1 2 3 1 2 + 4 5
We do not get the sum of the two matrices.
Linear Algebra
Mathematica cannot perform the indicated operation because m1 and m2 are not lists. Now we do it correctly. m1 = {{1, 1}, {1, 2}} m2 = {{2, 3}, {4, 5}} {{1, 1}, {1, 2}} {{2, 3}, {4, 5}} m1 + m2 //MatrixForm 3 4 5 7
Construct a 10 10 diagonal matrix whose diagonal entries are the first ten primes.
SOLUTION
primelist = Array[Prime, 10] {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} DiagonalMatrix[primelist] // MatrixForm 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 29
Prime is a built-in Mathematica function.
Construct a 5 5 upper triangular matrix of 1s with 0s below the main diagonal.
SOLUTION
m = Table[If[i < = j, 1, 0], {i, 5}, {j, 5}]; m // MatrixForm 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1
12.6 Construct a 7 7 tridiagonal matrix with 2s on the main diagonal, 1s on the diagonals adjacent to the main diagonal, and 0s elsewhere.
SOLUTION
m = Table[If[Abs[i j] 1, 1, If[i j, 2, 0]], {i, 1, 7}, {j, 1, 7}] ; m // MatrixForm 2 1 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2
Linear Algebra
Let M be the 6 6 matrix containing the integers 1 through 36. Construct a 3 3 matrix consisting of the elements in the odd rows and even columns of M.
SOLUTION
m = Table[6i + j, {i, 0, 5}, {j, 1, 6}]; m // MatrixForm 1 7 13 19 25 31 2 8 3 9 4 5 6 12 18 24 30 36
10 11
14 15 16 17 20 21 22 23 26 27 28 29 32 33 34 35
m[[{1, 3, 5}, {2, 4, 6}]] // MatrixForm 2 4 6 14 16 18 26 28 30
12.2 Matrix Operations
Since vectors and matrices are stored as lists in Mathematica, all list operations described in 3 apply. In addition, there are some specialized commands that are applicable specifically to matrices. Since n-dimensional vectors can be considered to be n 1 matrices, many of these commands apply to vectors as well. In the following descriptions, m, m1, and m2 denote matrices and v1 and v2 denote vectors.
m1 + m2 computes the sum of two matrices. m1 m2 computes the difference of two matrices. c m multiplies each element of m by the scalar c. m1. m2 computes the matrix product of m1 and m2. v1.v2 computes the dot product of v1 and v2. For matrices, the operation returns a list; for vectors, a single number is returned. Cross[v1, v2] or v1 v2 returns the cross product of v1 and v2. (This applies to threedimensional vectors only.) The cross product symbol, , can be inserted into the calculation by typing (without spaces) the key sequence [ESC]c-r-o-s-s[ESC] . (Do not confuse this with the on the Basic Math Input palette. The latter represents simple multiplication.)
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