**
MATH4141/MATH6651/PHYS5070**

**FW02**

**Assignment
4 Due date: Friday, Nov. 8**

1. Show that the following matrices are orthogonal:

(a) the elementary reflectorwhere;

(b) the rotation matrix **P**
which differs from the identity matrix in four elements, viz,:

p_{ii} = p_{jj}
= c

p_{ij} = -p_{ji}
= s

for i not equal to j and c^{2}
+ s^{2} = 1.

2. The **Jacobi method **uses
a rotation matrices of the form given in question 1(b) to carry out
similarity transformations of a symmetric matrix **A**, i.e. a
sequence of matrices are generated where **A**^{(i+1)}^{
}= **P**^{T}**A**^{(i)}**P** and **A**^{(1)}
= **A**.

(a) If **A´** =
**P**^{T}**AP** show that the elements of **A´**
are identical to those of **A** except for

a**´**_{ki}_{
}= ca_{ki}_{ }-_{ }sa_{kj} =
a**´**_{ik}

a**´**_{kj}_{
}= ca_{kj}_{ }+ sa_{ki} = a**´**_{jk}

for k not equal to i or j and

a**´**_{ii} =
c^{2}a_{ii} + s^{2}a_{jj} - 2csa_{ij}

a**´**_{jj} =
c^{2}a_{jj} + s^{2}a_{ii} + 2csa_{ij}

a**´**_{ij} =
(c^{2} - s^{2})a_{ij} + cs(a_{ii} -
a_{jj}) = a**´**_{ji}

where a_{mn}_{ }
are the elements of **A**. Note that **A´** is symmetric
if **A** is.

(b) The basis of the Jacobi
method is to choose c (and s) such that a**´**_{ij}
= 0. If S is the sum of the off diagonal elements of **A**, i.e.

S =

and
S**´** is similarly defined for **A´** show that
S**´** = S - 2|a_{ij}|^{2} when a**´**_{ij}_{
}= 0, using the results of part (a). Since

this shows that repeated Jacobi transformations for a sequence of (i,
j) values will result in a sequence of matrices whose off diagonal
elements tend to zero. The diagonal elements will then tend to the
eigenvalues of **A**.

(c) In order to have a**´**_{ij}_{
}= 0 show that c =_{ }[1 + t^{2}]^{-1/2},
s = t c and t can be chosen to be

3. The following IMSL routines implement the Housholder plus QR method to find all the eigenvalues and eigenvectors of a real, symmetric matrix:

i) imsl_d_eig_sym (C version);

ii) DEVCSF (double precision Fortran version).

Write a program to use one of these routines to find all the eigenvalues and eigenvectors of the following matrices:

(a) _{
};

(b) _{
};

(c) _{
}.

Format your output to display each of the eigenvalues with the corresponding eigenvector.

4.
If a matrix **A** has pseudoinverse **A**^{+} prove it
has the following properties:

i)
**AA**^{+}**A** = **A **

ii)
**A**^{+}**AA**^{+} = **A**^{+}

iii) (**AA**^{+})^{T}
= **AA**^{+}

iv) (**A**^{+}**A**)^{T}
= **A**^{+}**A**

Hint:
If the singular value decomposition of **A**^{ } is **UWV**^{T}
show first that **W** has these four properties. These properties
are known as the Penrose properties.

5. The amount of a product p of a chemical reaction present at time t is measured. The results are given in the following table:

t |
p |
---|---|

0.100 |
4.066 |

0.200 |
4.492 |

0.300 |
4.901 |

0.400 |
5.296 |

0.500 |
5.704 |

We want to find a linear least-squares fit to these data of the form ct + d = p where c and d are unknowns to be determined. If we put the data in the form of a system of linear equations

**Ax**
= **p **where A is a 5 by 2 matrix whose first column consists of
the values of t and the second column has all elements equal to one,
**x** is the vector of unknowns [c, d]^{T} and **p **is
the vector of p values then the least-squares solution is given by **A**^{+
}**p**. Use the IMSL routines DLSGRR or lin_svd_gen to find
the pseudoinverse of **A**. Use this to find the least squares
solution to the system of linear equations given above.