Dwigodsky_Assignment3

Data 605

1)Problem set 1

1) What is the rank of the given matrix?
Matrix 3 is rank 4

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

## [1] "Our final row echelon form is:"

##      [,1] [,2] [,3]       [,4]
## [1,]    1    2    3  4.0000000
## [2,]    0   -2   -4 -7.0000000
## [3,]    0    0    8  5.0000000
## [4,]    0    0    0 -0.3103448

2) For an m x n matrix, m > n , what is the maximum rank? What is the minimum rank, if it is non-zero?
The maximum rank is N. The minimum is 1. Like an algebra problem in k variables and k+1 equations, where one equation is unnecessary to fully specify a system, we can’t put more information in a matrix than its smallest dimension. Likewise, the least amount we could put in is one variable, unless it’s zero.

2)What is the rank of matrix B?
Matrix B is

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

It’s rank is one. Row 2 is 3 times row 1 and row 3 is twice row 1. These reduce down to 1 unique row.

1)Problem set 2

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

To find the eigenvectors, we find the determinant of (Lambda (I_N) - A). Because this is in row echelon form, the eigenvalues are just the numbers on the diagonals. Lambda - I~m,m , for all rows, is the only diagonal that doesn’t go to zero.

The characteristic polynomial is (x-1)(x-4)(x-6)

For eigenvalue 1:

## [1] "Subtract Matrix A from I*lambda"

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

## [1] "Put our new matrix in reduced row echelon form."

##      [,1] [,2]       [,3]
## [1,]    0    0  0.3333333
## [2,]    0   -3 -5.0000000
## [3,]    0    0 -5.0000000

##      [,1] [,2]       [,3]
## [1,]    0    0  0.3333333
## [2,]    0    3  0.0000000
## [3,]    0    0 -5.0000000

##      [,1] [,2]      [,3]
## [1,]    0    0 0.3333333
## [2,]    0    3 0.0000000
## [3,]    0    0 0.0000000

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

X₁ = 0, x₂ = 0, x₃ = any real

For eigenvalue 4:

## [1] "Subtract Matrix A from I*lambda"

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

## [1] "Put our new matrix in reduced row echelon form."

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

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

##      [,1]       [,2] [,3]
## [1,]    1 -0.6666667    0
## [2,]    0  0.0000000    1
## [3,]    0  0.0000000    0

X₂ = any real, x₁ = 2/3*x₁, x₃ = any real

For eigenvalue 6:

## [1] "Subtract Matrix A from I*lambda"

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

## [1] "Put our new matrix in reduced row echelon form."

##      [,1] [,2] [,3]
## [1,]    5    0   -8
## [2,]    0    2   -5
## [3,]    0    0    0

##      [,1] [,2] [,3]
## [1,]    1    0 -1.6
## [2,]    0    1 -2.5
## [3,]    0    0  0.0

X₃ = any real, x₁ = 1.6x₃, x₂ = 2.5 X₃