LECTURE 11-25-13
MATRICES
Matrices: a rectangular array of numbers that ovey certain rules of algebra.
An m x n matrix consists of m rows and n columns.
Example 1:
2 x 3 matrix:
a = 88-2, 1, 3, :l Ø I5 + 33 M>>
2 2
We can skip the secular equation by using the Eigenvalues command:

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l =

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1 1
: I5 + 33 M, I5 - 33 M>
2 2
The eigenvectors may be obtained using the Eigenvectors command:

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1 1
:: I-3 + 33 M, 1>, : I-3 - 33 M, 1>>
6 6
There is another command, Eigensystem, that gives the eigenvalues and eigenvectors all in one shot:

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1 1 1 1
:: I5 + 33 M, I5 - 33 M>, :: I-3 + 33 M, 1>, : I-3 - 33 M, 1>>>
2 2 6 6
Now let’s verify the similarity transform.
First, we store the eigenvectors as columns in the matrix x
NOTE:
they were given as rows, so we need to transpose the eigenvectors matrix returned by the Eigenvector
command
Printed by Wolfram Mathematica Student Edition 8 10 - matrices and eigens.nb
x =

[email protected]@aDD;

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1 1
6 I-3 + 33 M 6I-3 - 33 M
1 1
Here is the similarity transform:
xinv =

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xinv.a.x;

[email protected]%D êê MatrixForm
-16
5.37228 8.88178 µ 10
-17
4.16334 µ 10 -0.372281
Use Chop command to zero out the numbers that are effectively zero:

[email protected]%D êê MatrixForm
5.37228 0
K O
0 -0.372281
Compare to the diagonal matrix of eigenvalues:

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[email protected]%D êê MatrixForm
K 5.37228 0. O
0. -0.372281
IT WORKS!!
Now we do another example involving a 3 x 3 matrix:

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a = 881, 2, 3
r r
The general form of the volume element in a coordinate system defined by the coordinates u , u , u is
1 2 3
dV = h1h 2 3u d1 du2, w3ere h , 1 , 2 a3e “scale factors” that are particular to a given coordinate
system.
For example:
In the Cartesian coordinate system, the coordinates are1u = x,2u = y,3u = z, and the scale factors are:
Printed by Wolfram Mathematica Student Edition 10 10 - matrices and eigens.nb

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8h1, h2, h3< =

[email protected]"Cartesian", "ScaleFactors", 8x, y, z