MTH603 Grand Quiz Solution and Discussion

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues.
The eigenvectors of a symmetric matrix A corresponding to different eigenvalues are orthogonal to each other.

Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues.

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).
Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is nonzero since P is invertible) is an eigenvector for B with eigenvalue λ.

If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
Eigenvalues of a _________ matrix are all real.
If each entry of an n×n matrix A is a real number, then the eigenvalues of A are all real numbers. False. In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.

Eigenvalues of a _________ matrix are all real.

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
The Jacobi iteration converges, if A is strictly diagonally dominant.
If A is strictly row diagonally dominant, then the Jacobi iteration converges for any choice of the initial approximation x(0). However, the Jacobi iteration may converge for a matrix that is not strictly row diagonally dominant.

The Jacobi iteration converges, if A is strictly diagonally dominant.

The linear equation 2x+0y2=0 has solution(s).


Let[A]bea3×3realsymmetricmatrixwitha23bethenumericallylargestoff−diagonalelementthenusingJacobi′smethodthevalueofθcanbefoundby
Let A be a 3×3 matrix with real entries. Prove that if A is not similar over
R to a triangular matrix then A is similar over C to a diagonal matrix. 
While using Relaxation method, which of the following is increment ‘dxi’corresponding to the largest Residual for 1st iteration on the system;
2x+3y = 1, 3x +2y =  4 ? 
Central Difference method is the finite difference method
Finite difference. A finite difference is a mathematical expression of the form f (x + b) − f (x + a). … The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.


Whileusingtherelaxationmethodforfindingthesolutionofthefollowingsystem11x1+x2−x3=8 x1+8x2+5x3=9 x1+x2+9x3=7withtheinitialvector(0,0,0),theresidualswouldbe

By using determinants, we can easily check that the solution of the given system of linear equation ______ and it is ______.