Consider the matrix A below. Find the eigenvalues of A using a calculator and use these to construct the characteristic polynomial of A, pA (x). State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for A by computing expressions for null spaces, only using your calculator to row-reduce matrices. State the geometric multiplicity of each eigenvalue. Is A diagonalizable? If not, explain why. If so, find a diagonal matrix D that is similar to A.
library(pracma)## Warning: package 'pracma' was built under R version 3.4.3A <- matrix(c(19,-23,7,-3,25,-30,9,-4,30,-35,10,-5,5,-5,1,-1), ncol = 4)
A## [,1] [,2] [,3] [,4]
## [1,] 19 25 30 5
## [2,] -23 -30 -35 -5
## [3,] 7 9 10 1
## [4,] -3 -4 -5 -1eigen(A)## eigen() decomposition
## $values
## [1] -1.000000e+00 -1.000000e+00 -3.907985e-14 -6.416384e-16
##
## $vectors
## [,1] [,2] [,3] [,4]
## [1,] 0.6812872 -0.61709131 0.63660701 -0.705670095
## [2,] -0.7033974 0.74700527 -0.73209807 0.696145134
## [3,] 0.1539456 -0.22734943 0.22281245 -0.131609058
## [4,] -0.1318354 0.09743547 -0.09549105 -0.009524962I4 <- diag(4)
n1 <- nullspace(A + I4)
n0 <- nullspace(A)
S <- cbind(n1, n0)
S## [,1] [,2] [,3] [,4]
## [1,] -0.4197166 -0.75752094 -0.7696914 -0.2204557
## [2,] -0.5349477 0.62642025 0.3028420 0.6873058
## [3,] 0.6797882 -0.04662364 0.3129112 -0.5109412
## [4,] 0.2748762 0.17772432 -0.4668495 0.4668501D <- solve(S) %*% A %*% S
round(D, 3)## [,1] [,2] [,3] [,4]
## [1,] -1 0 0 0
## [2,] 0 -1 0 0
## [3,] 0 0 0 0
## [4,] 0 0 0 0A1 <- S %*% D %*% solve(S)
round(A1)## [,1] [,2] [,3] [,4]
## [1,] 19 25 30 5
## [2,] -23 -30 -35 -5
## [3,] 7 9 10 1
## [4,] -3 -4 -5 -1We see that A is also similar to D
The geometric and algebraic multiplicities of the eigenvalues 0, -1 are 2 and 2, respectively.
This matrix is daigonalizable because it forms a basis for C4, the dimension of the original matrix. (However it is not orthongonlly diagonalizable ;)