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For matrix \(A=\left[\begin{array} {rrr} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{array}\right]\) the characteristic polynomial of \(A\) is \(p_A(x)=(4-x)(1-x)^2\). Find the eigenvalues and corresponding eigenspaces of \(A\)
By definition _, the eigenvalues of a square matrix \(A\) are all the values of \(x\) that make the characteristic polynomial of \(A\) equal to \(0\). These are \(x=4, 1\). Thus the eigenvalues are \(\lambda = 4, 1\)
By definition _, the eigenspaces of a square matrix \(A\) is the null space of the following matrix:
\(\left[\begin{array} {rrr} 2-\lambda & 1 & 1 \\ 1 & 2-\lambda & 1 \\ 1 & 1 & 2-\lambda \\ \end{array}\right]\)
For \(\lambda = 4\):
\(\left[\begin{array}{rrr} 2-4 & 1 & 1 \\ 1 & 2-4 & 1 \\ 1 & 1 & 2-4 \\ \end{array}\right] = \left[\begin{array} {rrr} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \\ \end{array}\right]\)
pracma::rref(matrix(c(-2, 1, 1, 1, -2, 1, 1, 1, -2), 3, 3, byrow=TRUE))## [,1] [,2] [,3]
## [1,] 1 0 -1
## [2,] 0 1 -1
## [3,] 0 0 0The null space of this matrix is \(\Bigg \langle \left[\begin{array} {rrr} 1 \\ 1 \\ 1 \\ \end{array}\right] \Bigg \rangle\)
We repeat the process for each eigenvalue. \(\lambda = 1\):
\(\left[\begin{array} {rrr} 2-1 & 1 & 1 \\ 1 & 2-1 & 1 \\ 1 & 1 & 2-1 \\ \end{array}\right] = \left[\begin{array} {rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right]\)
pracma::rref(matrix(rep(1, 9), 3, 3, byrow=TRUE))## [,1] [,2] [,3]
## [1,] 1 1 1
## [2,] 0 0 0
## [3,] 0 0 0The null space of this matrix is \(\Bigg \langle \left[\begin{array} {rrr} -1 \\ 1 \\ 0 \\ \end{array}\right], \left[\begin{array} {rrr} -1\\ 0\\ 1\\ \end{array}\right] \Bigg \rangle\)
Summarizing:
\(E_\lambda(4) = N(A-4I_3)=\Bigg \langle \left[\begin{array} {rrr} 1 \\ 1 \\ 1 \\ \end{array}\right] \Bigg \rangle\)
\(E_\lambda(1) = N(A-1I_3)=\Bigg \langle \left[\begin{array} {rrr} 1 \\ -1 \\ 0 \\ \end{array}\right], \left[\begin{array} {rrr} 1\\ 0\\ -1\\ \end{array}\right] \Bigg \rangle\)