Eigenvalues and Eigenvectors
We will cover following topics
Characteristic Polynomial
The characteristic polynomial of a matrix \(A\) is formed by equationg \(det(A-\lambda I)\) to 0. It gives the eigen values of \(A\) and then corresponding to each eigen value, the eigen vector can be found as \(Ax=\lambda x\).
Eigenvalues and Eigenvectors
For a matrix \(A\), if \(Ax= \lambda x\), then \(\lambda\) is called an eigen vector of \(A\) corresponding to the eigen vector \(\lambda\).
To calculate eigen vectors and eigen values, we first form the characteristic polynomial \(\begin{vmatrix} A-\lambda I \end{vmatrix} = 0\) and then for each \(\lambda\), we find the corresponding eigen vector by solving \(Ax=\lambda x\)
Cayley-Hamilton Theorem
Cayley-Hamilton Theorem: It states that every square matrix satisfies its own characteristic equation \(\begin{vmatrix} A-\lambda I \end{vmatrix} = 0\).
PYQs
Characteristic Polynomial
1) Show that similar matrices have the same characteristic polynomial.
[2017, 10M]
2) If \(A=\begin{bmatrix}{1} & {1} & {3} \\ {5} & {2} & {6} \\ {-2} & {-1} & {-3}\end{bmatrix}\) then find \(A^{14}+3 A-2 I\).
[2016, 4M]
3) If \(\lambda\) is a characteristic root of a non-singular matrix \(A\), then prove that \(\dfrac{\vert A \vert }{\lambda}\) is a characteristic root of \(\operatorname{Adj} A\).
[2012, 8M]
4) Find the characteristic polynomial of the matrix \(A= \begin{bmatrix}{1} & {1} \\ {-1} & {3}\end{bmatrix}\). Hence, find \(A^{-1}\) and \(A^{6}\).
[2004, 15M]
5) A square matrix \(A\) is non-singular if and only if the constant term in its characteristic polynomial is different from zero.
[2002, 12M]
6) If \(\lambda\) is a characteristic root of a non-singular matrix \(A\) then prove that \(\dfrac{\vert A \vert}{\lambda}\) is a characteristic root of \(Adj(A)\).
[2001, 12M]
Eigenvalues and Eigenvectors
1) Show that if \(A\) and \(B\) are similar \(n\times n\) matrices, then they have the same eigenvalues.
[2018, 12M]
2) Prove that distance non-zero eigenvectors of a matrix are linearly independent.
[2017, 10M]
3) If \(A= \begin{bmatrix}{1} & {1} & {0} \\ {1} & {1} & {0} \\ {0} & {0} & {1}\end{bmatrix}\), then find the Eigen values and Eigenvectors of \(A\).
[2016, 6M]
4) Find the Eigen values and Eigen vectors of the matrix \(\begin{bmatrix}{1} & {1} & {3} \\ {1} & {5} & {1} \\ {3} & {1} & {1} \end{bmatrix}\).
[2015, 12M]
5) Let \(A= \begin{bmatrix}{-2} & {2} & {-3} \\ {2} & {1} & {-6} \\ {-1} & {-2} & {0}\end{bmatrix}\). Find the Eigen values of \(A\) and the corresponding Eigen vectors.
[2014, 8M]
6) Let \(A\) be a square matrix and \(A^{*}\) be its adjoint, show that the Eigen values of matrices \(A A^{*}\) and \(A^{*} A\) are real. Further show that \(\operatorname{trace}\left(A A^{*}\right)=\operatorname{trace}\left(A^{*} A\right)\).
[2013, 10M]
7) Let \(A= \begin{bmatrix}{1} & {1} & {1} \\ {1} & {\omega^{2}} & {\omega} \\ {1} & {\omega} & {\omega^{2}}\end{bmatrix}\), where \(\omega( \neq 1)\) is a cube root of unity. If \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\) denote the Eigen values of \(A^{2}\), show that \(\vert \lambda \vert_{1}+\vert \lambda_{2}\vert +\vert \lambda_{3}\vert \leq 9\).
[2013, 8M]
8) Let \(\lambda_{1}\), \(\lambda_{2}\), \(\ldots \ldots \lambda_{n}\) be the Eigen values of a \(n \times n\) square matrix \(A\) with corresponding Eigen vectors \(X_{1}, X_{2}, \ldots . X_{n}\). If \(B\) is a matrix similar to \(A\), show that the Eigen values of \(B\) are same as that of \(A\). Also, find the relation between the Eigen vectors of \(\mathrm{B}\) and Eigen vectors of \(\mathrm{A}\).
[2011, 10M]
9) Let \(A= \begin{bmatrix}{2} & {-2} & {2} \\ {1} & {1} & {1} \\ {1} & {3} & {-1}\end{bmatrix}\) and \(C\) be a non-singular matrix of order \(3 \times 3\). Find the Eigen values of the matrix \(B^{3}\) where \(B=C^{-1} A C\)
[2011, 10M]
10) If \(\lambda_{1}\), \(\lambda_{2}\), \(\ldots \ldots_{3}\) are the Eigen values of the matrix \(\mathrm{A}= \begin{bmatrix}{26} & {-2} & {2} \\ {2} & {21} & {4} \\ {44} & {2} & {28}\end{bmatrix}\), show that \(\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}} \leq \sqrt{1949}\)
[2010, 12M]
11) Let \(A\) and \(B\) be \(n \times n\) matrices over reals. Show that \(I-BA\) is invertible if \(I-A B\) is invertible. Deduce that \(A B\) and \(BA\) have the same Eigen values.
[2010, 20M]
12) Prove that the Eigen vectors corresponding to distinct Eigen values of a square matrix are linearly independent.
[2003, 15M]
13) Let \(A\) be a real \(3 \times 3\) symmetric matrix with Eigen values 0, 0 and 5. If the corresponding Eigen-vectors are (2,0,1), (2,1,1) and (1,0,-2), then find the matrix \(A\).
[2002, 15M]
Cayley-Hamilton Theorem
1) State the Cayley-Hamilton theorem. Use this theorem to find \(A^{100}\), where
\[A = \begin{bmatrix} 1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{bmatrix}\][15M]
2) If matrix \(A= \begin{bmatrix}{1} & {0} & {0} \\ {1} & {0} & {1} \\ {0} & {1} & {0}\end{bmatrix}\) then find \(A^{30}\).
[2015, 12M]
3) Verify the Cayley-Hamilton theorem for the matrix
\[\begin{bmatrix} 1 & 0 & -1 \\ 2 & 1 & 0 \\ 3 & -5 & 1 \end{bmatrix}\]Using this, show that A is non-sinugular and find A^{-1}.
[2011, 10M]
4) Verify Cayley-Hamilton theorem for the matrix \(A=\begin{bmatrix}{1} & {4} \\ {2} & {3}\end{bmatrix}\) and hence find its inverse. Also, find the matrix represented by \(A^{5}-4 A^{4} - 7A^3+11 A^{2}-A-10 I\).
[2014, 10M]
5) State Cayley-Hamilton theorem and using it, find the inverse of \(\begin{bmatrix}{1} & {3} \\ {2} & {4}\end{bmatrix}\).
[2006, 12M]
6) If \(A= \begin{bmatrix}{2} & {1} & {1} \\ {0} & {1} & {0} \\ {1} & {1} & {2}\end{bmatrix}\), then find the matrix represented by \(2 A^{10-} 10 A^{9}+14 A^{8}-6 A^{7}-3 A^{6}+15 A^{5}-21 A^{4}+9 A^{3}+A-1\).
[2003, 12M]
7) Use Cayley-Hamilton theorem to find the inverse of the following matrix: \(\begin{bmatrix}{0} & {1} & {2} \\ {1} & {2} & {3} \\ {3} & {1} & {1}\end{bmatrix}\).
[2002, 15M]
8) If \(A= \begin{bmatrix}{1} & {0} & {0} \\ {1} & {0} & {1} \\ {0} & {1} & {0}\end{bmatrix}\), show that for every integer \(n \geq 3\), \(A^{n}=A^{n-2}+A^{2}-I\). Hence, determine \(A^{50}\).
[2001, 15M]