Previous Year Questions on Eigen Values and Eigen Vectors

Eigen Values and Eigen Vectors - Previous Year Questions

Previous Year Questions on Eigen Values and Eigen Vectors

Previous year questions on Eigen Values and Eigen Vectors provide insights into their pivotal role in mathematics and science. Introduced during the 19th century, eigenvalues and eigenvectors became a cornerstone of Linear Algebra Notes. Their applications range from stability analysis in physics to principal component analysis in machine learning.

Vidyasagar University

2023-24 (CBCS)

If \( \lambda\) be an eigen value of an \(n \times n \) idempotent matrix \(A\), prove that \( \lambda\) is either \(1\) or \(0\). [2]
If \(A\) be a square matrix, then show that the sum of the characteristic roots of \(A\) is equal to the trace of \(A\). [2]
Find the eigen values and eigen vectors of \( A=\begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \). Also find the algebraic and geometric multiplicities of the eigen velues. [5]
Show that eigen values of a real symmetric matrix are all real. [4]

2021-22 (CBCS)

Use Cayley-Hamilton’s theorem to find \(A^{50} \), where \(A= \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}\) . [4]
Show that eigen values of the matrix \(A=\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix} \) are all real. [2]

2020-21 (CBCS)

If \(A\) and \(P\) be both \(n\times n\) matrices and \(P\) be non singular, then \(A\) and \(P^{-1}AP\) have the same eigen values. [2]
\(A\) and \(B\) are real orthogonal matrices of the same order and \(|A|+|B|=0 \). Show that \(A+B\) is a singular matrix. [2]
Find eigen value of a matrix of order \(n\). If \(\lambda \) be an eigen value of an \(n\times n\) idempotent matrix \(A\), then prove that \(\lambda\) is either \(1\) or \(0\). [2]

2019-20 (CBCS)

Define an eigen vectors of a matrix \(A_{n\times n} \) over a field \(F\). Show that there exists many eigen vectors of \(A\) corresponding to an eigen value \(\lambda\in F\). [2]
State Cayley-Hamilton’s theorem and use it to find \(A^{100} \), where \( A= \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}\) . [1+4]
If \(S\) be a real skew symmetric matrix of order \(n\), prove that
1. the matrix \(S-I_{n}\) is non-singular
2. the matrix \(\left(S-I_{n}\right)^{-1}\left(S+I_{n}\right)\) is orthogonal
3. the matrix \(S-I_{n}\) is non-singular
4. if \(X\) be an eigen vector of \(S\) with eigen value \(\lambda\), then \(X\) is also an eigen vector of \(\left(S-I_{n}\right)^{-1}\left(S+I_{n}\right)\) with eigen values \(\frac{\lambda+1}{\lambda-1}\)
. [1+2+3]

2018-19 (CBCS)

\(A\) and \(B\) are any two \(2\times 2\) matrices and \(E\) is the corresponding unit matrix. Show that \(AB-BA=E\) cannot hold any circumstances.
If \( \lambda\) be an eigen value of an non-singular matrix \(A\), then prove that \( \lambda^{-1}\) is an eigen value \( A^{-1}\). [2]
If \(A= \begin{pmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix}\) then show that \(A^{2}=A^{-1}\). [2]
\(A\) is \(3\times 3\) real matrix having the eigen values \(2,3,1\). If \( \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \( \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\), \( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) are the eigen vectors of A corresponding to the eigen values \(2,3,1\) respectively. Find the matrix \(A\). [2+2+1]
Prove that eigen values of a real skew symmetric matrix are purely imaginary or zero. [4]

2017-18 (CBCS)

State Cayley-Hamilton’s theorem and using theorem find \(A^{-1} \), where \( A=\begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix}\). [2]
Verify Cayley-Hamilton’s theorem for the matrix \( A=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 1 \\ 2 & 3 & 2 \end{pmatrix}\). Hence compute \(A^{-1} \). [3+2]
If \(X_{1},X_{2},…,X_{r}\) be \(r\) eigen vectors of an \(n\times n \) matrix \(A\) corresponding to \(r\) distinct eigen values \(\lambda_{1},\lambda_{2},…,\lambda_{r}\) respectively, then prove that \(X_{1},X_{2},…,X_{r}\) are linearly independent. [5]
\(\lambda\) is an eigen value of a real skew symmetric matrix. Prove that \(\left| \frac{1-\lambda}{1+\lambda} \right|=1 \). [2]

FAQs

What are eigenvalues and eigenvectors?
Eigenvalues are scalars, and eigenvectors are directions that remain unchanged under a linear transformation.
Why are eigenvalues and eigenvectors important?
They are critical for simplifying complex systems, such as stability analysis and data reduction techniques.
What is the characteristic equation?
It is a polynomial equation derived from the determinant of a matrix, used to find eigenvalues.
What is diagonalization?
It is a process of expressing a matrix in a diagonal form using its eigenvalues and eigenvectors.
Where are eigenvalues used in real life?
They are used in vibration analysis, quantum mechanics, and image compression.
What is the geometric meaning of eigenvectors?
Eigenvectors represent directions in which a transformation acts by stretching or compressing.
What is spectral decomposition?
It is the representation of a matrix using its eigenvalues and eigenvectors.
How are eigenvalues calculated?
They are found by solving the determinant of \( (A – \lambda I) \) = 0, where \(A\) is the matrix and \(\lambda \) is the eigenvalue.
What is the difference between eigenvalues and singular values?
Eigenvalues pertain to square matrices, while singular values apply to any matrix via singular value decomposition.
Can all matrices have eigenvalues?
Not all matrices have real eigenvalues; some may have complex eigenvalues.