1. Problem set 1

  1. What is the rank of the matrix A?

\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ -1 & 0 & 1 & 3 \\ 0 & 1 & -2 & 1 \\ 5 & 4 & -2 & -3 \end{bmatrix} \]

Using row reduction: the rank is the number of non-zero rows after row reduction performed.

rankMatrix(A)
## [1] 4
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 8.881784e-16

Using row reduction: the rank is the number of non-zero rows after row reduction performed. Rank matrix A = 4

  1. Given an mxn matrix where m > n, what can be the maximum rank? The minimum rank, assuming that the matrix is non-zero?

The maximum rank can be equal or less than n (minimum of m and n), while the minimum rank for non-zero matrix is 1.

  1. What is the rank of matrix B?
B <- matrix(c(1,3,2,2,6,4,1,3,2), ncol = 3, nrow = 3)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
rankMatrix(B)
## [1] 1
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 6.661338e-16

Rank matrix B = 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.

\[ C = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} \]

\[ det ( \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} - \begin{bmatrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ \end{bmatrix} ) = 0 \]

\[ det ( \begin{bmatrix} 1-λ & 2 & 3 \\ 0 & 4-λ & 5 \\ 0 & 0 & 6-λ \end{bmatrix} ) = 0 \]

(1-λ)(4-λ)(6-λ) = 0

(1-λ)(24 - 10λ + λ2) = 0

λ3 - 11λ2 +34λ - 24 = 0

Eigenvalues:

λ1 = 6

λ2 = 1

λ3 = 4

Eigenvectors:

  1. λ1 = 6

\[ det ( \begin{bmatrix} 1-λ & 2 & 3 \\ 0 & 4-λ & 5 \\ 0 & 0 & 6-λ \end{bmatrix} ) = \begin{bmatrix} -5 & 2 & 3 \\ 0 & -2 & 5 \\ 0 & 0 & 0 \end{bmatrix} = 0 \]

v = (1, 1.6, 2.5)

1.6/sqrt((1.6)^2+(2.5)^2+(1)^2)
## [1] 0.5108407
2.5/sqrt((1.6)^2+(2.5)^2+(1)^2)
## [1] 0.7981886
1/sqrt((1.6)^2+(2.5)^2+(1)^2)
## [1] 0.3192754
  1. λ2 = 1

\[ det ( \begin{bmatrix} 1-λ & 2 & 3 \\ 0 & 4-λ & 5 \\ 0 & 0 & 6-λ \end{bmatrix} ) = \begin{bmatrix} 0 & 2 & 3 \\ 0 & 3 & 5 \\ 0 & 0 & 5 \end{bmatrix} = 0 \]

v = (0, 0, 1)

1/sqrt(1^2)
## [1] 1
0/sqrt(1^2)
## [1] 0
0/sqrt(1^2)
## [1] 0
  1. λ3 = 4

\[ det ( \begin{bmatrix} 1-λ & 2 & 3 \\ 0 & 4-λ & 5 \\ 0 & 0 & 6-λ \end{bmatrix} ) = \begin{bmatrix} -3 & 2 & 3 \\ 0 & 8 & 5 \\ 0 & 0 & 10 \end{bmatrix} = 0 \]

v = (0, 2/3, 1)

1/sqrt((1)^2+(2/3)^2)
## [1] 0.8320503
2/3/sqrt((1)^2+(2/3)^2)
## [1] 0.5547002
0/sqrt((1)^2+(2/3)^2)
## [1] 0

Checking results (eigenvalues and eigenvectors) with eigen().

C <- matrix(c(1, 0, 0, 2, 4, 0, 3, 5, 6), 3, 3)
eigen(C)
## 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