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UW / Applied Mathematics / AMATH 301 / What is the difference between matrix operation and component operatio

What is the difference between matrix operation and component operatio

What is the difference between matrix operation and component operatio

Description

School: University of Washington
Department: Applied Mathematics
Course: Beginning Scientific Computing
Professor: Ulrich hetmaniuk
Term: Spring 2016
Tags: Matlab, Linear Algebra, and Linear Systems
Cost: 25
Name: AMATH 301 Week 2 Notes: Solving Linear Systems
Description: Notes from week 2: Norms (definitions and different types) and techniques of solving linear systems (Ax=b) through Gaussian elimination, LU decomposition, Matlab "\" command.
Uploaded: 04/11/2016
13 Pages 72 Views 4 Unlocks
Reviews


Matrix Operation  vs. Component Operations

Example: We also discuss several other topics like What does central dogma do?

1.>> A = [ 1, 3, 3, 5, 6, 7, 8, 3, 1 ];

     A ^ 2   % A² = A.A  ( matrix multiplication) If you want to learn more check out How can one calculate relative prices and what do they mean?

   

      Ans: 32      27       25Don't forget about the age old question of What does prenatal care include?

              91      72       59

              31      45       38Don't forget about the age old question of What is the concept of arousal theory?
If you want to learn more check out What is the conecept of durkheim's theory?

2. A ^ 2    %  squares each component

     Ans: 1      9       4Don't forget about the age old question of What is the concept of james lange's theory of emotion in psychology?

            25    36      44

            64     9        1

3. A./A     % divides each component by corresponding component in A

 

     Ans:   1      1      1

               1      1      1

               1      1      1

4. A/A  % outputs solution to Ax = A (Output: x)

     Ans:    1   0     0

                0   1     0

                0   0     1

Max command:

[ max_vector, location] = max (A)

max _vector = 8 6 7 - max element in each column

Location: = 3 2 2 - where in column max appears

>> [ max_vector2 , location 2] = max (A, [   ], 2) finds max in each row

>> [ max_whole mat, location 3 ] = max (A (:)) finds mox in whole matrix

→ A (:) outputs A as a vector


Norms

A generalized definition at distance → different types of norms, different types of distance

Ex. Euclidean Distance (2-norm)

1- norm (Taxi-cab/ Manhattan Distance)

∞- norm ( infinity/ max-norm)

  Norms in MotLab

>> b = [ I, o ,-4 ]

1 >> norm (b) % Motlab default is 2-norm, norm (b) is same as norm (b,2)

     Ans: 4,1231

2>> norm (b, 1)

     Ans: 5

3>> norm (b, int) % in Motlab int is ∞

    Ans: 4

4>> max labs (b)  % abs (b) is lbl (absolute value)

   Ans: 4

-Note: norm (b, inf)= max (abs (b))

Complex Numbers

>>c = [ 3 + 4 x i, 2, 3 ,5 x i ]

1>> norm (c, int)

  Ans: 5

2>>norm (c, 2)

 Ans: 6.3

3>> Mox (c) % 5 i is longest element

 Ans: 5 i

4>> mox (abs (c) % 15 il = 5

 Ans: 5

Matrix Norms

Ex. >> A = [ 1, 3, 2, 2, 0, -5, 8, -2, 1 ]

1>> Norm (A , 1 )

  Ans: 11

2>> Norm (A, int)

  Ans: 11

Matrix 2 - norm

Given matrix A, how does it act on a unit circle. Vectors that define unit circle are rotated and scaled.

IIAII2 = length of longest semi- axis

>> norm (A, 2 ) % in Motlab, matrix norm is called same way as vector norm

Ans = 8.4787

Solving Linear System

Eq 1:  2x1  +   4x2  -  2x3   =2

Eq 2:  4x1  +   9x2  -  3x3 = 8

Eq 3: -2x1    -   3x2   -  7x3 = 10

Then back substitute, yet x3= 2, x2, x1 = -1

In Matrix Form

Same as eq. Form takes linear combination of rows of A

Advantage in given new b, just need to compute L⁻¹, already computed L and V, which is the expensive part. This makes it faster and cheaper than Gaussian elimination, which has to do the entire process every time.

     

     

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