install.packages(“pracma”)

Problem Set 1

(1) What is the rank of the matrix A?

\[\mathbf{A} = \left[\begin{array} {} 1 & 2 & 3 & 4\\ -1 & 0 & 1 & 3\\ 0 & 1 & -2 & 1\\ 5 & 4 & -2 & -3\\ \end{array}\right] \]

A <- matrix(c(1, 2, 3, 4, -1, 0, 1, 3, 0, 1, -2, 1, 5, 4, -2, -3), nrow = 4, ncol = 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
y <- qr(A)

rank <- y$rank

rank
## [1] 4
The rank of A is 4.
(2) Given an mxn matrix where m > n, what can be the maximum rank? The minimum rank, assuming that the matrix is non-zero?
If A is m x n, it follows the inequalities that

\[\\rank(A_{max})\leq min(m , n) \]

Therefore, the maximum rank is n, and the minimum rank is 1.
(1) What is the rank of the matrix B?

\[\mathbf{B} = \left[\begin{array} {} 1 & 2 & 1\\ 3 & 6 & 3\\ 2 & 4 & 2\\ \end{array}\right] \]

B <- matrix(c(1, 2, 1, 3, 6, 3, 2, 4, 2), nrow = 3, ncol = 3, byrow = TRUE)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
z <- qr(B)

rank <- z$rank

rank
## [1] 1
The rank of B is 1.

Problem Set 2

\[\mathbf{A} = \left[\begin{array} {} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 6\\ \end{array}\right] \]

library(pracma)
A <- matrix(c( 1, 2, 3, 0, 4, 5, 0, 0, 6), nrow = 3, ncol = 3, byrow = TRUE)
A
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
p <- charpoly(A, info = TRUE)
## Error term: 0
p$cp
## [1]   1 -11  34 -24
Therefore, the characteristic polynomial is

\[\lambda^3\ - 11\lambda^2\ + 34\lambda\ - 24 = 0\]

ev <- eigen(A)
values <- ev$values
values
## [1] 6 4 1

\[\lambda = 6, 4,1 \]

vec <- ev$vectors
vec
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0

\[when\quad \lambda = 6, \]

\[\mathbf span ({E_{\lambda = 6}} = \left[\begin{array} {} 0.5108407 \\ 0.7981886 \\ 0.3192754 \end{array}\right]) \]

\[when\quad \lambda = 4, \]

\[\mathbf span ({E_{\lambda = 4}} = \left[\begin{array} {} 0.5547002 \\ 0.8320503 \\ 0 \end{array}\right]) \]

\[when\quad \lambda = 1, \]

\[\mathbf span ({E_{\lambda = 1}} = \left[\begin{array} {} 1 \\ 0 \\ 0 \end{array}\right]) \]