CHAPTER 2: MATRIX COMPUTATIONS AND LINEAR ALGEBRA
Lecture 2.3: MATLAB linear
algebra
Linear systems of m equations for n
unknowns:
a11 x1 +
a12 x2 + … + a1n xn = b1
a21 x1 +
a22 x2 + … + a2n xn = b2
. . .
am1 x1 +
am2 x2 + … + amn xn = bm
·
m = n: a square system with a square matrix A (it has typically a
unique solution)
·
m < n: underdetermined system (it has typically infinitely many solutions)
·
m > n: overdetermined system (it has typically no solutions)
Example:
A = 
, b = 
. The unique solution exists: x1 = 0.2, x2
= -0.8.
MATLAB operations and functions:
·
"/", "\": matrix divisions (solutions of a linear system)
A = [ 3 2 ; 1, -1]; b = [-1 ; 1];
x = A \ b % the column-oriented solution of a linear system
dif = A*x – b % check
that the solution x is correct
-0.8000
dif = 0
0
AA = [3 1;
2,-1]; bb = [-1,1]; % transposed matrices A and b
xx = bb/AA % the row-oriented solution of a linear system
dif = xx*AA-bb % check that the solution x is correct
dif = 0 0
· inv: matrix inversion (B is the inverse of A if B*A = I)
A = [ 3 2 ; 1, -1]; B = inv(A), BB = A^(-1) % two equivalent operations
I1 = B*A, I2 = A*B % check that B is the inverse of A
0.2000 -0.6000
BB = 0.2000
0.4000
0.2000 -0.6000
I1 = 1.0000 -0.0000
0
1.0000
I2 = 1.0000
0
-0.0000 1.0000
·
det: computes
the matrix determinant (det A = a11*a22 – a12*a21
for a 2-by-2 matrix A)
·
trace: computes the sum of the
diagonal elements of a matrix (the trace)
·
rref: computes
the row reduced echelon form of a matrix
·
rank: computes
the number of linearly independent rows or columns of a matrix (the rank)
·
null: computes
a basis of eiegenvectors for the homogeneous equation A*x = 0
·
eig: computes
eigenvalues and eigenvectors of the linear problem A*x = 
*x
D = det(A), T = trace(A), R = rank(A)
T = 2
R = 2
% Alternative computations of solutions of
linear systems via A^(-1)
A = [ 3 2 ; 1, -1]; b = [-1 ; 1];
x = inv(A)*b % it is less efficient because of larger computational time
-0.8000
% Alternative
computations of solutions by using the Cramer's rule
A1 = [ b, A(:,2) ], A2 =
[ A(:,1), b], clear x
x(1) = det(A1)/det(A), x(2) = det(A2)/det(A)
1 -1
A2 = 3 -1
1 1
x = 0.2000
x = 0.2000 -0.8000
% Alternative
computations of solutions by using the augmented matrix [A,b]
Rrow = rref([A,b])
0 1.0000 -0.8000
% D: diagonal matrix of eigenvalues
% V: fundamental matrix of eigenvectors
% V^(-1)*A*V = D: diagonalization formula
Dif = V^(-1)*A*V - D
0.2193 0.9121
D = 3.4495 0
0 -1.4495
Dif = 1.0e-015 *
-0.8882 0.2220
0.2776 0
N = null(A) % null-space is the basis of eigenvectors for zero eigenvalues
Empty matrix: 2-by-0
Types of solutions
of linear systems:
·
A
linear square system of linear equations has a unique solution iff det(A)
0.
·
When
det(A) = 0, the linear system may have infinitely many solutions
or no solutions at all.
·
If
det(A) 
0, the inverse matrix A^(-1) exists and
A is called a non-singular matrix.
·
If
det(A) = 0, inv(A) does not exist and A is called a
singular matrix. The MATLAB linear algebra solvers A\b or b/A
do not produce any valuable solution, if A is singular.
A1 = [ -1,1; -2,2]; b1 = [1 ; 0];
% the system is
inconsistent and has no solution
x1 = A1\b1, B1 = inv(A1), D1 = det(A1)
Warning: Matrix is singular to working precision.
x1 = Inf
Inf
Warning: Matrix is singular to working precision.
B1 = Inf Inf
Inf Inf
D1 = 0
A2 = [ -1,1; -2,2]; b2 = [1 ; 2];
% the second equation is
redundant
% the system has infinitely
many solutions: x(2) = 1 + x(1)
x2 = A2\b2
A3 = [ -1, 1 ]; b3 = [ 1
]; % the second equation is removed
x3 = A3\b3
Warning: Matrix is singular to working precision.
x2 = Inf
Inf
x3 = -1
0
% comparison of properties of non-singular
matrix A versus singular matrix A1
S1 = rref(A) % non-singular
matrix has the identity matrix in RREF
S2 = rref(A1) % a singular
matrix has zeros in one or more rows in RREF
0 1
S2 = 1 -1
0 0
R1 = rank(A1) % singular matrices have rank smaller than 2
N1 = null(A1) % the null-space of singular matrices is non-empty
[V1,D1] = eig(A1) % singular matrices have zero eigenvalues
N1 = 0.7071
0.7071
V1 = -0.7071 -0.4472
-0.7071 -0.8944
D1 = 0 0
0 1
Rrow1 = rref([A2,b2]) % accurate solution of the system with singular matrices
0 0 0