1. Engineering Analysis ENG 3420 Fall 2009
Dan C. Marinescu
Office: HEC 439 B
Office hours: Tu-Th 11:00-12:00

2. Lecture 17
Reading assignment Chapters 10 and 11, Linear Algebra ClassNotes
Last time:
Symmetric matrices; Hermitian matrices.
Matrix multiplication
Today:
Linear algebra functions in Matlab
The inverse of a matrix
Vector products
Tensor algebra
Characteristic equation, eigenvectors, eigenvalues
Norm
Matrix condition number
Next Time
More on
LU Factorization
Cholesky decomposition
Lecture 17 2

3. Matrix analysis in MATLAB
Norm  Matrix or vector norm
normest  Estimate the matrix 2-norm
rank  Matrix rank
det  Determinant
trace  Sum of diagonal elements
null  Null space
orth  Orthogonalization
rref  Reduced row echelon form
subspace Angle between two subspaces

4. Eigenvalues and singular values
eig  Eigenvalues and eigenvectors
svd  Singular value decomposition
eigs  A few eigenvalues
svds  A few singular values
poly  Characteristic polynomial
polyeig  Polynomial eigenvalue problem
condeig  Condition number for eigenvalues
hess  Hessenberg form
qz  QZ factorization
schur  Schur decomposition

5. Matrix functions Expm  Matrix exponential Logm  Matrix logarithm
Sqrtm  Matrix square root
Funm  Evaluate general matrix function

6. Linear systems of equations
\ and /  Linear equation solution
inv  Matrix inverse
cond  Condition number for inversion
condest  1-norm condition number estimate
chol  Cholesky factorization
cholinc  Incomplete Cholesky factorization
linsolve  Solve a system of linear equations
lu  LU factorization
ilu  Incomplete LU factorization
luinc  Incomplete LU factorization
qr  Orthogonal-triangular decomposition
lsqnonneg  Nonnegative least-squares
pinv  Pseudoinverse
lscov Least squares with known covariance

7. The inverse of a square
If [A] is a square matrix, there is another matrix [A]-1, called the inverse of [A], for which [A][A]-1=[A]-1[A]=[I]
The inverse can be computed in a column by column fashion by generating solutions with unit vectors as the right-hand-side constants:

8. Canonical base of an n-dimensional vector space
100……000
010……000
001……000
…………….
000…….100
000…….010
000…….001

9. Matrix Inverse (cont)
LU factorization can be used to efficiently evaluate a system for multiple right-hand-side vectors - thus, it is ideal for evaluating the multiple unit vectors needed to compute the inverse.

10. The response of a linear system
The response of a linear system to some stimuli can be found using the matrix inverse.

11. Distance and norms
Metric space   a set where the ”distance” between elements of the set is defined, e.g., the 3-dimensional Euclidean space. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.
A norm  real-valued function that provides a measure of the size or “length” of an element of a vector space.

12. Vector Norms The p-norm of a vector X is:
Important examples of vector p-norms include:

13. Matrix Norms Common matrix norms for a matrix [A] include:
Note - max is the largest eigenvalue of [A]T[A].

14. Matrix Condition Number
The matrix condition number Cond[A] is obtained by calculating Cond[A]=||A||·||A-1||
In can be shown that:
The relative error of the norm of the computed solution can be as large as the relative error of the norm of the coefficients of [A] multiplied by the condition number.
If the coefficients of [A] are known to t digit precision, the solution [X] may be valid to only t-log10(Cond[A]) digits.

15. Built-in functions to compute norms and condition numbers
norm(X,p)  Compute the p norm of vector X, where p can be any number, inf, or ‘fro’ (for the Euclidean norm)
norm(A,p)  Compute a norm of matrix A, where p can be 1, 2, inf, or ‘fro’ (for the Frobenius norm)
cond(X,p) or cond(A,p)  Calculate the condition number of vector X or matrix A using the norm specified by p.

16. LU Factorization LU factorization involves two steps:
Decompose the [A] matrix into a product of:
a lower triangular matrix [L] with 1 for each entry on the diagonal.
and an upper triangular matrix [U
Substitution to solve for {x}
Gauss elimination can be implemented using LU factorization
The forward-elimination step of Gauss elimination comprises the bulk of the computational effort.
LU factorization methods separate the time-consuming elimination of the matrix [A] from the manipulations of the right-hand-side [b].

17. Gauss Elimination as LU Factorization
To solve [A]{x}={b}, first decompose [A] to get [L][U]{x}={b}
MATLAB’s lu function can be used to generate the [L] and [U] matrices: [L, U] = lu(A)
Step 1  solve [L]{y}={b}; {y} can be found using forward substitution.
Step 2 solve [U]{x}={y}, {x} can be found using backward substitution.
In MATLAB: [L, U] = lu(A) d = L\b x = U\d
LU factorization  requires the same number of floating point operations (flops) as for Gauss elimination.
Advantage  once [A] is decomposed, the same [L] and [U] can be used for multiple {b} vectors.

18. Cholesky Factorization
A symmetric matrix  a square matrix, A, that is equal to its transpose:
A = AT (T stands for transpose).
The Cholesky factorization  based on the fact that a symmetric matrix can be decomposed as:
[A]= [U]T[U]
The rest of the process is similar to LU decomposition and Gauss elimination, except only one matrix, [U], needs to be stored.
Cholesky factorization with the built-in chol command: U = chol(A)
MATLAB’s left division operator \ examines the system to see which method will most efficiently solve the problem. This includes trying banded solvers, back and forward substitutions, Cholesky factorization for symmetric systems. If these do not work and the system is square, Gauss elimination with partial pivoting is used.