Dat 605 HW3 - Eigenvalues & Eigenvectors

library(matlib)
library(MASS)
library(pracma)

Prob 1

• What is rank of matrix A, using rref R’s built-in function

https://www.rdocumentation.org/packages/pracma/versions/1.9.9/topics/rref

A1 = matrix(c(1,2,3,4,-1,0,1,3,0,1,-2,1,5,4,-2,-3), nrow = 4, ncol = 4, byrow=T)

A1

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

rref(A1) 

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

Ans:

• Since after row-reduction, we get an Identity 4x4 matrix (4 linearly independent columns of vectors or 4 non-zero pivots)
• The rank of A1 is 4

---

• Given an mxn matrix where m > n, what is the:

1. max rank
2. min rank

Ans:

• For non-zero rectangular matrices, the rank has to be no greater than the smaller of the row or column dimension

Therefore, max rank is n and the min rank is n

---

• What is rank of matrix B, using rref R’s built-in function again to solve

B = matrix(c(1,2,3,0,4,5,0,0,6), nrow = 3, ncol = 3, byrow=T)

B

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

rref(B) 

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1

---

Ans:

• Since after row-reduction, we get an Identity 3X3 matrix (3 linearly independent columns of vectors or 3 non-zero pivots)
• The rank of B is 3

---

Prob 2

Compute Eigenvalues and Eigenvectors of matrix A:

Fig 1a: Eigenvalues

Fig 1b: Eigenvectors: