# Finite Element Method Questions and Answers – Dynamic Considerations – Evaluation of Eigen Values & Eigen Vectors

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Dynamic Considerations – Evaluation of Eigen Values & Eigen Vectors”.

1. Eigenvectors possess the property of being orthogonal with respect to the stiffness matrix only.
a) True
b) False

Explanation: The given statement is false. Eigenvectors possess the property of being orthogonal with respect to both the stiffness and mass matrices respectively. It is mathematically represented as –
UiTMUj = 0; i! = j
UiTKUj = 0; i! = j
where M and K are the mass and stiffness matrices respectively.

2. What are the different types of categories of eigenvalue-eigenvector evaluation?
a) Characteristic polynomial, Varied Iteration and Transformation
b) Characteristic polynomial, Varied Iteration and Translation
c) Characteristic polynomial, Vector Iteration and Transformation
d) Characteristic polynomial, Varied Iteration and Deformation

Explanation: There are three different categories of eigenvalue-eigenvector evaluation. They are Characteristic polynomial method, Vector Iteration method and Transformation method respectively. Vector Iteration is further branched into Inverse integration, power integration method etc.

3. What is the generalized expression for the characteristic polynomial method?
a) (K – λM)U = 0
b) (K + λM)U = 0
c) (K * λM)U = 0
d) (K / λM)U = 0

Explanation: The generalized expression for the characteristic polynomial method is given as (K – λM)U = 0. Here K and M stand for stiffness and mass matrices respectively, λ stands for eigenvalue and U stands for eigenvector. In case, the eigenvector is not trivial, the expression changes to det(K – λM) = 0

4. Which properties are explicitly used for the vector integration method?
a) Euler quotient
b) Newton quotient
c) Edison quotient
d) Rayleigh quotient

Explanation: Most integration methods make use of the properties of the Rayleigh quotient. It is represented mathematically as –
Q(v) = vTKV / vTMV, where v is the arbitrary vector
The value of Rayleigh’s quotient lies between the smallest and largest eigenvalue.

5. Which are the methods utilized for calculating the largest and smallest eigenvalues respectively?
a) Power, Inverse iteration
b) Power, Subspace iteration
c) Subspace, inverse iteration

Explanation: Vector iteration methods are best suited for calculating the largest and smallest eigenvalues. Power iteration method helps to arrive at the largest eigenvalues; whereas, the inverse iteration method helps evaluate the lowest eigenvalues. Subspace method is used to evaluate in large scale problems.
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6. What is the principle of the Gram – Schmidt method?
a) Any orthogonal set is non linearly independent
b) Any orthogonal set is linearly independent
c) Any orthogonal set is non linearly dependent
d) Any orthogonal set is linearly dependent

Explanation: The Gram – Schmidt method states that, all orthogonal sets are linearly independent. This has three main assumptions and is used for ortho-normalizing a set of vectors in the given product space.

7. What are the different types of transformation methods?
a) Ramsay, Eric method
b) Newton, Edison method
c) Jacobi, QR method
d) Euler, Rayleigh method

Explanation: There are two types of transformation methods – Jacobi method and QR method. These are suited for applications in large scale problems. In QR method, the matrices are reduced to traditional form using householder matrices. In Jacobi method, transformations are used to simultaneously diagonalize stiffness and mass matrices.

8. Which of the following is a property of householder transformation?
a) It is irrational
b) It is rational
c) It is not symmetric
d) It is symmetric

Explanation: Householder transformation has two important properties. They are, symmetry transformation and inverse property. The householder transformation is symmetric in nature and its inverse is itself.

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