CARNEGIE MELLON UNIVERSITY
CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
18-771 LINEAR SYSTEMS SPRING 2003
PROBLEM SET II
200 POINTS
DUE 13 FEBRUARY 2003
in LECTURE
LINEAR ALGEBRA
MATRIX ALGEBRA
MATRICES: PROPERTIES AND APPLICATIONS
NORMS
LINEAR ALGEBRAIC EQUATIONS
LEAST SQUARES
THE ALGEBRAIC EIGENVALUE PROBLEM
EIGENVALUES AND EIGENVECTORS
SINGULAR VALUES
RANK
CONDITIONING
FUNCTIONS OF MATRICES
MATLAB( COMPUTATION IN ENGINEERING
READING ASSIGNMENT
WEEK B
Week of 27 January: 3
Week of 3 February: 4
Week of 10 February: 5
PROBLEM B PAGE
VI * 3.17 and 3.19 143 25 POINTS
PROBLEM VII *
25 POINTS
A) Compute manually and with MATLAB( the eigenvalues, eigenvectors and singular values of the [3 x 3] circular shift matrix
[pic]
(B) Interpret and generalize your findings for [n x n] circular shift matrices.
PROBLEM VIII *
25 POINTS
Compute manually and with MATLAB( as a function of the parameter p the singular values and condition number of the matrix
[pic]
where j = +sqrt(-1).
PROBLEM IX *
75 POINTS
In the system of linear algebraic equations
A x = y (1)
the coefficient matrix
[pic]
and the right-hand side input vector
[pic]
A) Apply MATLAB( to compute all of the solutions of (1).
B) Compute the solutions of (1) produced by applying the MATLAB( commands
x = A\y, x = pinv (A) * y
and rref. Interpret your findings.
PROBLEM X *
50 POINTS
The matrix-vector oriented MATLAB® is the premier software system for computer-aided engineering analysis and design. The backslash operator \ pervades numerical linear algebra computation in MATLAB® for solving the system of linear algebraic equations
A x = b (1)
In (1), A(mxn) is the rank r coefficient matrix, x(nx1) is the solution vector and b(mx1) is the right-hand side input vector.
Explain in detail and illustrate diagrammatically (in flow chart format) what MATLAB® returns in response to the backslash operator command
x = A \ b
for m>n, m=n and m ................
................
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