Changing the MatrixSo far, only discussed changing the RHS, i.e. Ax = b ! Abx = bb.The matrix consists of FP numbers, too—it, too, is approximate. I.e.

Ax = b ! bAbx = b.

What can we say about the error due to an approximate matrix?

58

Changing Condition NumbersOnce we have a matrix A in a linear system Ax = b, are we stuck with itscondition number? Or could we improve it?

What is this called as a general concept?

59

In-Class Activity: Matrix Norms and Conditioning

In-class activity: Matrix Norms and Conditioning

60

Singular Value Decomposition (SVD)

What is the Singular Value Decomposition of an m ⇥ n matrix?

61

Computing the 2-Norm

Using the SVD of A, identify the 2-norm.

Express the matrix condition number cond2(A) in terms of the SVD:

62

Not a matrix norm: FrobeniusThe 2-norm is very costly to compute. Can we make something simpler?

What about its properties?

63

Frobenius Norm: Properties

Is the Frobenius norm induced by any vector norm?

How does it relate to the SVD?

64

* Errata: the square root was missing from theoriginal scribbles

Solving Systems: Simple casesSolve Dx = b if D is diagonal. (Computational cost?)

Solve Qx = b if Q is orthogonal. (Computational cost?)

Given SVD A = UΣV T , solve Ax = b. (Computational cost?)

65

Solving Systems: Triangular matricesSolve

a11 a12 a13 a14a22 a23 a24

a33 a34a44

xyzw

=

b1b2b3b4

.

Demo: Coding back-substitution [cleared]What about non-triangular matrices?

66

Gaussian Elimination

Demo: Vanilla Gaussian Elimination [cleared]What do we get by doing Gaussian Elimination?

How is that different from being upper triangular?

What if we do not just eliminate downward but also upward?

67

LU Factorization

What is the LU factorization?

68

Solving Ax = b

Does LU help solve Ax = b?

69

Determining an LU factorization

Demo: LU Factorization [cleared]70

Computational Cost

What is the computational cost of multiplying two n ⇥ n matrices?

I u11 = a11, uT12 = aT

12.I l 21 = a21/u11.I L22U22 = A22 − l 21uT

12.

What is the computational cost of carrying out LU factorization on ann ⇥ n matrix?

Demo: Complexity of Mat-Mat multiplication and LU [cleared]

71