(7.3)
REVISION OF MATRICES
GROUP I
Consider the matrices A and B:
|
|
A = |
2 3 0 1 |
|
B = |
1 1 0 3 |
|
|
|
1 1 4 1 |
|
|
1 3 2 5 |
|
|
|
0 4 2 2 |
|
|
2 1 6 0 |
|
|
|
1 0 3 3 |
|
|
2 2 1 0 |
1. Using a spreadsheet, calculate: A + B, A * B, Det(A), Det(B), A-1 e B-1
2. Show that (A.B)-1 = B-1.A-1
3. Show that (A.B)T = BT.AT
GROUP II
Let the Matrices:
|
M(4,4) = (1/4) |
1 1 1 1 |
|
O(4,4) = |
0 0 0 0 |
|
I(4,4) = |
1 0 0 0 |
|
|
1 1 1 1 |
|
|
0 0 0 0 |
|
|
0 1 0 0 |
|
|
1 1 1 1 |
|
|
0 0 0 0 |
|
|
0 0 1 0 |
|
|
1 1 1 1 |
|
|
0 0 0 0 |
|
|
0 0 0 1 |
1. Verify that the matrix null 0 is idempotent.
2. Verify that the matrix identity I is idempotent.
3. Verify that the matrix M is idempotent.
4. What are the traces of M and I?
5. Calculate the ranks, r, of M and of I.
6. What is the value of the determinant of M and I?
GROUP III
1. Verify that the product Mx, where x is the vector given by
xT = (3 4 8 1), is a vector with all elements equal to the arithmetic
mean, 
, of the 4 elements of
vector x.
2. Verify that (I-M)x is the vector of the deviation.
3. Verify that the sum of the squares of xi, Σ(xi2) can be written as: xT. x
4. Verify that the sum of the squares of the deviations,
, can also be written in a
matricial form, as: xT (I-M)x
GROUP IV
|
1. Consider the vector |
x = 2 + θ |
where θ is an unknown parameter. |
|
|
3 θ |
|
a) Write the derivative
of the vector x
b) Calculate xT x
c) Calculate
d) Show that
|
2. Consider the vector x = |
2 + 4θ1 - 5θ2 |
where θ1 and θ2 are two unknown constants. |
|
|
1+ θ1 + θ2 |
|
a) Write the derivative matrix
(take θ1 and θ2 as variables)
b) Calculate xT x
c) Transpose
d) Show that the transposed matrix
GROUP V
Consider the following system of 2 equations with 2 unknowns;
5 = 2 A + 3 B
4 = A - 2 B
1. Show that the equation system can be written in matrix form as,
Y(2,1) = X(2,2) θ(2,1)
where Y is the vector of the independent terms (5 e 4) of the system,
θ is the vector of the unknowns A and B
and X is the matrix of the coefficients of the unknowns
2. Verify that the solution of the system can be given as θ = (XTX)-1XTY
3. Show that X is a square, non singular matrix, and then that the solution of the system can be θ = X-1Y
ESTIMATION OF THE PARAMETERS OF THE YOSHIMOTO AND CLARKE MODEL (1993)
4. Estimate the parameters k, q and r, of the Fox integrated model (IFOX) and of Yoshimoto & Clarke (1993) using the following data:
|
Year |
Y |
CPUE |
|
1983 |
538 |
235 |
|
1984 |
638 |
131 |
|
1985 |
431 |
63 |
|
1986 |
99 |
22 |
|
1987 |
37 |
8 |
|
1988 |
62 |
21 |
|
1989 |
437 |
77 |
|
1990 |
146 |
28 |
|
1991 |
126 |
26 |
|
1992 |
53 |
25 |
|
1993 |
91 |
41 |
|
1994 |
232 |
66 |
which represent the total annual catches (in tons) and the respective catches by fishing effort unit (kg/fishing day of the fleet PESCRUL) of the stock of Deepwater rose shrimp, Parapenaeus longirostris of the Algarve during the period 1983 to 1994 (Mattos Silva, 1995).
Comment on the obtained results comparing them with those presented in Section 8.20.
+4θ2
