Linear Algebra: Eigen Values, Eigen Vectors, Cayley-Hamilton theorem Problems
Keywords : Eigenvalues, Eigenvectors, Cayley-Hamilton theorem, Diagonalizable, Hermitian matrix, unitary matrix, orthogonal matrix.
(1) For each matrix, find all eigenvalues and the corresponding linearly independent eigenvectors:
(a) \( \left[\begin{array}{lll} 3 & 1 & 1 \\ 2 & 4 & 2 \\ 1 & 1 & 3 \end{array}\right]^{\prime} \)
(b) \( \left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \)
(c) \( \left[\begin{array}{lll} 1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4 \end{array}\right] \)
(d) \( \left[\begin{array}{ccc} 7 & 2 & -2 \\ -6 & -1 & 2 \\ 6 & 2 & -1 \end{array}\right] \)
(2) (a) Let \( \lambda \) be an eigenvalue of a nonsingular square matrix \( A \) of order \( n \) and \( x \) be the corresponding eigenvector. Show that \( \lambda^{-1} \) is an eigenvalue of \( A^{-1} \) and identify the corresponding eigenvector. Also, identify the eigenvalue and eigenvector of \( A – k I \), where \( I \) is the identity matrix and \( k \) is a scalar.
(b) Let \( A \) be a square matrix of size \( n \). Show that \( A \) and \( A^T \) have the same eigenvalues. Are their eigenvectors also the same?
(c) If \( A \) and \( P \) are square matrices of order \( n \), and \( P \) is nonsingular, then prove that \( A \) and \( P^{-1} A P \) have the same eigenvalues.
(3) Prove that:
(a) All eigenvalues of a Hermitian matrix are real.
(b) Eigenvalues of a skew-Hermitian matrix are purely imaginary or zero.
(c) Eigenvectors corresponding to two distinct eigenvalues of a real symmetric matrix are orthogonal.
(d) Eigenvalues of a unitary matrix have unit modulus.
(e) Any skew-symmetric matrix of odd order has zero determinant.
(f) The eigenvalues of an idempotent matrix are either 0 or 1.
(g) All eigenvalues of a nilpotent matrix are 0.
(4) Let \( A \) and \( B \) be square matrices of order \( n \). Show that if \( \lambda \) is an eigenvalue of \( AB \), then it will be an eigenvalue of \( BA \). Hence, prove that \( I – AB \) is invertible if and only if \( I – BA \) is invertible.
(5) Prove that every Hermitian matrix can be written as \( A + i B \), where \( A \) is a real symmetric matrix, and \( B \) is a real skew-symmetric matrix.
(6) Given that \( A = \left[\begin{array}{cc} 0 & 1 + 2i \\ -1 + 2i & 0 \end{array}\right] \), show that \( (I – A)(I + A)^{-1} \) is a unitary matrix.
(7) (a) The eigenvalues of a \( 3 \times 3 \) matrix \( A \) are \( 2, 2, 4 \), and the corresponding eigenvectors are \( (-2, 1, 0)^T, (-1, 0, 1)^T \), and \( (1, 0, 1)^T \). Find \( A \).
(b) Find a matrix \( P \) such that \( P^{-1} A P \) is a diagonal matrix, where \( A \) is:
(i) \( \left[\begin{array}{ccc} 4 & -2 & 0 \\ -2 & 2 & -2 \\ 0 & -2 & 4 \end{array}\right] \)
(ii) \( \left[\begin{array}{ccc} 0 & 2 & -1 \\ 2 & 3 & -2 \\ -1 & -2 & 0 \end{array}\right] \)
(iii) \( \left[\begin{array}{ccc} 2 & -2 & 3 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{array}\right] \)
(iv) \( \left[\begin{array}{ccc} 2 & 0 & -1 \\ 3 & 4 & 0 \\ 3 & 2 & 0 \end{array}\right] \)
(8) Find \( e^{2 A} \) and \( A^{50} \) when:
(a) \( A = \left[\begin{array}{ccc} -4 & 1 & 0 \\ 0 & -3 & 1 \\ 0 & 0 & -2 \end{array}\right] \)
(b) \( A = \left[\begin{array}{ccc} -2 & 4 & 3 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right] \)
(9) Using Cayley-Hamilton theorem, find the inverse of the following matrices:
(a) \( \left[\begin{array}{ccc} 1 & 3 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{array}\right] \)
(b) \( \left[\begin{array}{ccc} 7 & 6 & 2 \\ -1 & 2 & 4 \\ 3 & 6 & 8 \end{array}\right] \)
(c) \( \left[\begin{array}{ccc} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{array}\right] \)
(10) Verify Cayley-Hamilton theorem for the matrix \( A = \left[\begin{array}{ccc} 1 & 2 & 0 \\ -1 & 1 & 1 \\ 4 & 0 & 1 \end{array}\right] \). Hence, find \( A^{-1} \) and \( A^4 \).
(11) Let \( A = \left[\begin{array}{ccc} 4 & \alpha & -1 \\ 2 & 5 & \beta \\ 1 & 1 & \gamma \end{array}\right] \). Given that the eigenvalues of the matrix \( A \) are \( 3, 3, \delta \) (where \( \delta \neq 3 \)) and \( A \) is diagonalizable, find the values of constants \( \alpha, \beta, \gamma, \delta \).
(12) Find an orthogonal or unitary matrix \( P \) such that \( P^{*} A P \) is diagonal where \( A \) is given by the following:
(a) \( \left[\begin{array}{cc} 7 & 3 \\ 3 & -1 \end{array}\right] \)
(b) \( \left[\begin{array}{ccc} 11 & -8 & 4 \\ -8 & -1 & -2 \\ 4 & -2 & -4 \end{array}\right] \)
(c) \( \left[\begin{array}{cc} 1 & 1 – i \\ 1 + i & 2 \end{array}\right] \)
(13) Let \( A \) be a \( 5 \times 5 \) invertible matrix with row sums 1. That is \( \sum_{j=1}^5 a_{ij} = 1 \) for \( 1 \leq i \leq 5 \). Then, prove that the sum of all the entries of \( A^{-1} \) is 5.
(14) Let \( A \) be a nilpotent matrix. Show that \( I + A \) is invertible.
(15) Suppose that \( A^{15} = 0 \). Show that there exists a unitary matrix \( U \) such that \( U^* A U \) is \( 5 \times 5 \) upper triangular with diagonal entries 0.
Answers
1. (a) Eigenvalues: \( 2, 2, 6 \)
Eigenvectors: \((-1,0,1)^T\), \((-1,1,0)^T\), \((1,2,1)^T\)
(b) Eigenvalues: \(1, 1, 1\)
Eigenvectors: \((1,0,1)^T\), \((1,0,0)^T\)
(c) Eigenvalues: \( -2, -2, 4 \)
Eigenvectors: \( (0,1,1)^T\), \( (1,0,-1)^T \), \( (1,1,2)^T \)
(d) Eigenvalues: \( 1, 1, 3 \)
Eigenvectors: \( (1,0,3)^T\), \( (1,-3,0)^T\), \((-1,1,-1)^T\)
7. (a) \( A = \left[\begin{array}{lll} 3 & 2 & 1 \\ 0 & 2 & 0 \\ 1 & 2 & 3 \end{array}\right] \)
(b) (i) \( A = \left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 0 & -1 \\ 1 & 1 & 1 \end{array}\right] \)
(ii) \( A = \left[\begin{array}{ccc} 2 & 1 & -1 \\ -1 & 0 & -2 \\ 0 & 1 & 1 \end{array}\right] \)
(iii) \( A = \left[\begin{array}{ccc} -1 & -11 & 1 \\ 1 & -1 & 1 \\ 1 & 14 & 1 \end{array}\right] \)
(iv) \( A = \left[\begin{array}{ccc} 1 & -2 & -1 \\ -1 & 3 & 3 \\ 1 & 0 & 1 \end{array}\right] \)
8. (a) \( e^{2A} = \left[\begin{array}{ccc} \alpha & \beta-\alpha & \frac{\alpha – 2\beta + \gamma}{2} \\ 0 & \beta & \gamma – \beta \\ 0 & 0 & \gamma \end{array}\right] \)
where \( \alpha = e^{-8}, \beta = e^{-6}, \gamma = e^{-4} \) for \( e^{2A} \) and \( \alpha = 4^{50}, \beta = 3^{50}, \gamma = 2^{50} \) for \( A^{50} \).
(b) \( e^{2A} = \frac{1}{6} \left[\begin{array}{ccc} 6\alpha & 4(\beta-\alpha) & 3(\beta-\alpha) \\ 0 & 6\beta & 0 \\ 0 & 0 & 6\beta \end{array}\right] \)
where \( \alpha = e^{-4}, \beta = e^8 \) for \( e^{2A} \) and \( \alpha = 2^{50}, \beta = 4^{50} \) for \( A^{50} \).
9. Using Cayley-Hamilton theorem, the inverses of the matrices are:
(a) \( A^{-1} = \left[\begin{array}{ccc} 7 & -3 & -3 \\ -1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right] \)
(b) \( A^{-1} = \left[\begin{array}{ccc} -0.20 & -0.90 & 0.50 \\ 0.50 & 1.25 & -0.75 \\ -0.30 & -0.60 & 0.50 \end{array}\right] \)
(c) \( A^{-1} = \left[\begin{array}{ccc} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{array}\right] \)
10. \( A^{-1} = \frac{1}{11} \left[\begin{array}{ccc} 1 & -2 & 2 \\ 5 & 1 & -1 \\ -4 & 8 & 3 \end{array}\right] \)
\( A^4 = \left[\begin{array}{ccc} 25 & 8 & 8 \\ 12 & 25 & 4 \\ 16 & 32 & 33 \end{array}\right] \)
11. The values of the constants are:
\( \alpha = 1, \beta = -2, \gamma = 2, \delta = 5 \).
12. The orthogonal or unitary matrices are:
(a) \( P = \frac{1}{\sqrt{10}} \left[\begin{array}{cc} -1 & 3 \\ 3 & 1 \end{array}\right] \)
(b) \( P = \left[\begin{array}{ccc} 0 & \frac{-5}{\sqrt{105}} & \frac{4}{\sqrt{21}} \\ \frac{1}{\sqrt{5}} & \frac{-8}{\sqrt{105}} & \frac{-2}{\sqrt{21}} \\ \frac{2}{\sqrt{5}} & \frac{4}{\sqrt{105}} & \frac{1}{\sqrt{21}} \end{array}\right] \)
(c) \( P = \left[\begin{array}{cc} \frac{-2}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\ \frac{1+i}{\sqrt{6}} & \frac{1+i}{\sqrt{3}} \end{array}\right] \)