DATA 605 ASSIGNMENT 3

ASSIGNMENT 3 IS 605 FUNDAMENTALS OF COMPUTATIONAL MATHEMATICS - 2014

1. Problem set 1

1. What is the rank of the matrix A?

library(pracma)
A <- matrix(
    c(
        1,2,3,4,
        -1,0,1,3,
        0,1,-2,1,
        5,4,-2,-3
    )
    ,
    4,4,
    byrow = TRUE
)
A

##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]   -1    0    1    3
## [3,]    0    1   -2    1
## [4,]    5    4   -2   -3

Rank(A)

## [1] 4

Answer: Rank of matrix A is 4.

2. Given an mxn matrix where m > n, what can be the maximum rank? The mini- mum rank, assuming that the matrix is non-zero? Answer: The maximum rank is equal to the number of non-zero rows in its row echelon matrix i.e. equal to m. The minimum rank is one.
3. What is the rank of matrix B?

B <- matrix(
    c(1,2,1,
      3,6,3,
      2,4,2),
    3,3,
    byrow = TRUE
)
B

##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2

Rank(B)

## [1] 1

Answer: Rank of B is 1.

2. Problem set 2

Compute the eigenvalues and eigenvectors of the matrix A. You’ll need to show your work. You’ll need to write out the characteristic polynomial and show your solution. Answer:

A <- matrix(c(
    1,2,3,
    0,4,5,
    0,0,6
), 3, 3 , byrow = TRUE)
A

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6

# install.packages("matlib")
library(matlib)

## 
## Attaching package: 'matlib'

## The following objects are masked from 'package:pracma':
## 
##     angle, inv

Eigen Value of A

ev <- eigen(A)
ev$values

## [1] 6 4 1

Eigenvector of A

ev$vectors

##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0

Characteristic polynomial

Ap <- charpoly(A, info = TRUE)

## Error term: 0

Ap$cp

## [1]   1 -11  34 -24