library(Matrix)

Problem set 1

1) Rank of matrix A

A <- matrix(c(1 ,2, 3 ,4,-1, 0 ,1, 3,0 ,1, -2, 1,5 ,4 ,-2 ,-3), nrow = 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
paste("Rank of matrix A is ",rankMatrix(A))
## [1] "Rank of matrix 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?

Maximum rank = n (Maximum rank is the minimum of m and n) Minimum rank of a non-zero matrix = 1

3) Rank of matrix B

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

B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
paste("Rank of matrix B is ",rankMatrix(B))
## [1] "Rank of matrix B is  1"

Problem set 2

4) Eigenvalues and eigenvectors

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

\(|A - λI| = 0 = det\left[\begin{array}{ccc} 1-λ & 2 & 3\\ 0 & 4-λ & 5\\ 0 & 0 & 6-λ \end{array}\right]\)

Thereby our expected eigenvalues are 1,4,6. We can verify the same and also compute the eigenvectors as below

A<-matrix(c(1,0,0,2,4,0,3,5,6), nrow =3)
eigen(A)
## eigen() decomposition
## $values
## [1] 6 4 1
## 
## $vectors
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0