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Smutek to uczucie, jak gdyby się tonęło, jak gdyby grzebano cię w ziemi.
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Smutek to uczucie, jak gdyby się tonęło, jak gdyby grzebano cię w ziemi.

E = [ones(size(t)) exp(-t)]
E =
1.0000 1.0000
1.0000 0.7408
1.0000 0.4493
1.0000 0.3329
1.0000 0.2019
1.0000 0.1003
11-16
Solving Linear Systems of Equations
The least squares solution is found with the backslash operator.
c = E\y
c =
0.4760
0.3413
In other words, the least squares fit to the data is
– t
y( t) ≈ 0.4760 + 0.3413 e
The following statements evaluate the model at regularly spaced increments in
t, and then plot the result, together with the original data.
T = (0:0.1:2.5)';
Y = [ones(size(T)) exp(-T)]*c;
plot(T,Y,'-',t,y,'o')
You can see that E*c is not exactly equal to y, but that the difference might well
be less than measurement errors in the original data.
A rectangular matrix A is rank deficient if it does not have linearly independent
columns. If A is rank deficient, the least squares solution to AX = B is not
unique. The backslash operator, A\B, issues a warning if A is rank deficient and
produces a basic solution that has as few nonzero elements as possible.
11-17
11 Matrices and Linear Algebra
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.50
0.5
1
1.5
2
2.5
Underdetermined Systems
Underdetermined linear systems involve more unknowns than equations.
When they are accompanied by additional constraints, they are the purview of
linear programming. By itself, the backslash operator deals only with the
unconstrained system. The solution is never unique. MATLAB finds a basic
solution, which has at most m nonzero components, but even this may not be
unique. The particular solution actually computed is determined by the QR
factorization with column pivoting (see a later section on the QR factorization).
11-18
Solving Linear Systems of Equations
Here is a small, random example.
R = fix(10*rand(2,4))
R =
6 8 7 3
3 5 4 1
b = fix(10*rand(2,1))
b =
1
2
The linear system Rx = b involves two equations in four unknowns. Since the
coefficient matrix contains small integers, it is appropriate to use the format
command to display the solution in rational format. The particular solution is
obtained with
format rat
p = R\b
p =
0
5/7
0
-11/7
One of the nonzero components is p(2) because R(:,2) is the column of R with
largest norm. The other nonzero component is p(4) because R(:,4) dominates
after R(:,2) is eliminated.
The complete solution to the overdetermined (overdetermined?) system can be
characterized by adding an arbitrary vector from the null space, which can be
found using the null function with an option requesting a “rational” basis.
Z = null(R,'r')
Z =
-1/2 -7/6
-1/2 1/2
11-19
11 Matrices and Linear Algebra
1 0
0 1
It can be confirmed that R*Z is zero and that any vector x where
x = p + Z*q
for an arbitrary vector q satisfies R*x = b.
11-20
Inverses and Determinants
Inverses and Determinants
This section provides:
• An overview of the use of inverses and determinants for solving square
nonsingular systems of linear equations
• A discussion of the Moore-Penrose pseudoinverse for solving rectangular
systems of linear equations
Overview
If A is square and nonsingular, the equations AX = I and XA = I have the same
solution, X. This solution is called the inverse of A, is denoted by A-1, and is computed by the function inv. The determinant of a matrix is useful in
theoretical considerations and some types of symbolic computation, but its
scaling and roundoff error properties make it far less satisfactory for numeric
computation. Nevertheless, the function det computes the determinant of a
square matrix.
A = pascal(3)
A =
1 1 1
1 2 3
1 3 6
d = det(A)
X = inv(A)
d =
1
X =
3
-3 1
-3 5
-2
1
-2 1
Again, because A is symmetric, has integer elements, and has determinant
equal to one, so does its inverse. On the other hand,
B = magic(3)
11-21
11 Matrices and Linear Algebra
B =
8 1 6
3 5 7
4 9 2
d = det(B)
X = inv(B)
d =
-360
X =
0.1472
-0.1444 0.0639
-0.0611 0.0222 0.1056
-0.0194 0.1889
-0.1028
Closer examination of the elements of X, or use of format rat, would reveal
that they are integers divided by 360.
If A is square and nonsingular, then without roundoff error, X = inv(A)*B
would theoretically be the same as X = A\B and Y = B*inv(A) would
theoretically be the same as Y = B/A. But the computations involving the
backslash and slash operators are preferable because they require less
computer time, less memory, and have better error detection properties.
Pseudoinverses
Rectangular matrices do not have inverses or determinants. At least one of the
equations AX = I and XA = I does not have a solution. A partial replacement for
the inverse is provided by the Moore-Penrose pseudoinverse, which is computed
by the pinv function.
X = pinv(C)
X =
0.1159
-0.0729 0.0171
-0.0534 0.1152 0.0418
The matrix
Q = X*C
11-22
Inverses and Determinants
Q =
1.0000 0.0000
0.0000 1.0000
is the 2-by-2 identity, but the matrix
P = C*X
P =
0.8293
-0.1958 0.3213
-0.1958 0.7754 0.3685
0.3213 0.3685 0.3952
is not the 3-by-3 identity. However, P acts like an identity on a portion of the
space in the sense that P is symmetric, P*C is equal to C and X*P is equal to X.
If A is m-by- n with m > n and full rank n, then each of the three statements x = A\b
x = pinv(A)*b
x = inv(A'*A)*A'*b
theoretically computes the same least squares solution x, although the
backslash operator does it faster.
However, if A does not have full rank, the solution to the least squares problem
is not unique. There are many vectors x that minimize
norm(A*x -b)
The solution computed by x = A\b is a basic solution; it has at most r nonzero
components, where r is the rank of A. The solution computed by x = pinv(A)*b
is the minimal norm solution because it minimizes norm(x). An attempt to