Matlab:  Linear Algebra
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Practical Exercises
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Session 2 (Numerical Linear Algebra)
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2.1
    Set the matrix A to the following value:
       -0.2762    0.0060    0.4593
        0.2513    0.1991    0.0472
       -0.2449    0.3909   -0.3614
    Create a row vector B with the following value:
        0.4049   -0.3889    0.0949
    And a column vector C with the following value:
       -0.3152   -0.0611   -0.0913

    Solve the equations B = A*x and C = A*y, and multiply the results
by A to check that you get back to B and C.

    Print the condition number and the LAPACK approximate condition
number of A.


2.2
    Solve the equation B = A*x, and multiply the result by A to check
that you get back to B.  Note the errors.

    Print the condition number, the LAPACK approximate condition number
and eigenvalues of A.  Note the large values and the large number of
small eigenvalues.


2.3
    Set the matrix C to the following value:
        0.2127   -0.4576
        0.0005   -0.4286
       -0.0289    0.0216
       -0.4404   -0.4033
        0.1820    0.3181
    Set the matrix D to the following value:
       -0.3501    0.1596    0.0186    0.4730    0.1490

    Solve the equation D = C*x.


2.4
    There are no answers to this question, and I suggest leaving it
until later.  However, unless you are a matrix expert, I advise doing it
before using either the '/' or '\' operators with two matrix arguments.
Alternatively, avoid doing that.

    Create two 3x3 random matrices A and B, and investigate the
relationships between linsolve(x,y), x/y and x\y, with x and y being
either A and B or B and A.  Use the formulae given in the lecture as a
starting point.  The objective is to get a firm understanding of exactly
what the '/' or '\' operators actually do on matrices.


2.5
    Create a vector E of length 16 with the Nth element having value
cos(pi*(N-1)/4).
    
    Set F to be its Fast Fourier Transform.  Print F and the inverse
transform of F.


2.6
    Create an unsymmetric 5x5 matrix G by [G,y] = eig(hilb(5)).

    Print its 2-D fast Fourier transform using fftn, and then using fft.
Do the latter by row and column, and then by column and row.  All of
those should be the same, and there are other variations using
transposition.


2.7
    Set the matrix A to the following value:
       -0.2762    0.0060    0.4593
        0.2513    0.1991    0.0472
       -0.2449    0.3909   -0.3614

    Set B to the exponential of Z, and C to that of -Z.  Print B, C and
B times C and C times B.

    Calculate eye(3)+A+A^2/2+A^3/6+A^4/24+A^5/120 and note the slight
differences from B.

    Calculate the trace of A, both by using the trace function and by
summing the diagonal.


2.8
    Using the matrix A above, calculate D to be A times A transpose.
[Note that D is now symmetric ]

    Print D's 2-norm and determinant.

    Print its eigenvalues, their product, the singular values and their
product.  Note the relationships to each other, and to the norm and
determinant.

    Print its characteristic polynomial.


2.9
    Using the matrix A above (i.e. the original input), print its 2-norm
and determinant.

    Print its eigenvalues, their product, and the singular values.  Note
the relationships to each other, and to the norm and determinant.  [Note
that A is not normal ]

    Print its 1-norm, infinity-norm and eigenvalue condition numbers.


2.10
    Using the matrix D above, print its eigenvectors and singular
value decomposition.  Note the relationships (i.e. they are the same
results, rearranged).

    Use the relationships given in the lecture to recreate the matrix D
from the eigenvalues and eigenvectors, and the singular value
decomposition.

 
2.11
   Using the matrix A above, print its eigenvectors and singular value
decomposition.  Note the relationships, and that they differ.  [Note
that I said A was not normal. ]

    Use the relationships given in the lecture to recreate the matrix D
from the eigenvalues and eigenvectors, and the singular value
decomposition.