Rank EigenValues EigenVectors

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library(knitr)

Problem Set 1

What is the rank of the matrix A?

Answer: I’ll first code the matrix A, in the following code chunk.

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

First we observe that A is a square matrix, of order 4 x 4.

Now, we’ll compute the determinant of A, to check whether it’s singular or non-singular.

det(A)

## [1] -9

We observe that the determinant of A is -9, which is not 0.
Therefore A is a non-singular matrix.
Therefore, the rank of A is equal to its order, which 4.

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

Answer (first part of (2)): The maximum possible rank of the matrix is n.

Here’s the explanation (what I’ll state below, without proof, is variously formulated, in various books):

Definition of row-rank of a matrix is: “The maximum number of linearly independent rows of a matrix”.
Definition of column-rank of a matrix is: “The maximum number of linearly independent columns of a matrix”.

So, although the matrix could have m linearly independent rows, it could not have more than n independent columns. Therefore, while its row-rank could be m, its column-rank could not exceed n.

Now, row-rank of a matrix is equal to its column-rank. There is/are theorem(s) to prove this assertion. So, the rank of a matrix boils down to row-rank or column-rank, which are equivalent.

Therefore, if m > n, in no way could the rank of the matrix exceed n.

Foot note[1]: There is another definition of rank, which I used in problem (1) above, i.e.:
Rank of a matrix is the order of largest non-singular cofactor of the matrix.

And there are theorems to establish the equivalence of the two definitions.

Answer (second part of (2)): The minimum rank of a non-zero matrix can be 1.

Here’s the simple reason. Any cell of a matrix is itself a cofactor of order 1 x 1. If the matrix is non-zero, then it has at least one non-zero element. We can form a cofactor (of order 1), with that non-zero element. The determinant of that cofactor will be non-zero. So, minimum the rank can be 1.

What is the rank of matrix B?

Answer: I’ll first code the matrix B, in the following code chunk.

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

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

First we observe that B is a square matrix, of order 3 x 3.

Our next observation is that 1st and 3rd columns of B are identical, and that the 2nd column is the sum of 1st and 3rd columns.
Therefore, they are not linearly independent. So, B is non-singular. So, the first fact is rank of B is less than 3.

Now, from Theorem DRCMA (page 358), we know that the rank of a matrix remains unchanged, when the scalar multiple of a row (or column) is added to another row (or column).

So, with the operations (-3).R1 + R2, and (-2).R1 + R3, we get below row reduced matrix (B_Reduced1):

B_Reduced1 <- matrix(c(1, 0, 0, 2, 0, 0, 1, 0, 0), nrow = 3)
print(B_Reduced1)

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

Now, let’s do the operations (-2).C1 + C2 and (-1).C1 + C3 and get below matrix, which is further reduced (B_Reduced2).

B_Reduced2 <- matrix(c(1, 0, 0, 0, 0, 0, 0, 0, 0), nrow = 3)
print(B_Reduced2)

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

In B_Reduced2, the only surviving non-zero cofactor contains the element 1, in the cell B_Reduced2[1, 1]. The determinant of that cofactor is obviously 1 (not 0).

So, the rank of B must be 1.

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.

Answer: The Eigenvalues and Eigenvectors are computed, using Word’s Equation editor. After doing the math, screen shot of the word doc was written on to a ProblemSet_1.png file, which is inserted below:

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