How does matlab normalize its eigenvectors?

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Melissa il 3 Lug 2013

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https://it.mathworks.com/matlabcentral/answers/81024-how-does-matlab-normalize-its-eigenvectors

Good Afternoon All,

I am currently working with a mass-stiffness problem where I have two matrices M and K. Solving the eigenvalue problem I find the natural frequencies and the modeshapenatural frequencies with nastran as well as kinetic energy distribution (based off the modeshapes or eigenvectors) but my eigenvectors are not matching up. I was just wondering if anyone knew what matlab does in normalizing the eigenvectors?

I have also applied another transformation matrix to the K and M matrices and again receive the correct natural frequencies but my eigenvectors don't match either...hm.

If anyone could help explain I would greatly appreciate it.

Thanks,

Here is my code for producing natural frequencies and modeshapes:

K=[1200000, 0, 0, 0, 137797.6, -17335.2; 0, 1200000, 0, -137797.6, 0, -156678.4; 0, 0, 1200000, 17335.2, 156678.4, 0; 0, -137797.6, 17335.2, 97608.59896, -454.17925, 42493.43822; 137797.6, 0, 156678.4, -454.17925, 223401.7905, 1880.660199; -17335.2, -156678.4, 0, 42493.43822, 1880.660199, 282511.7668];
M=[300, 0, 0, 0, 0, 0; 0, 300, 0, 0, 0, 0; 0, 0, 300, 0, 0, 0; 0, 0, 0, 10.6, -1.419, -7.017; 0, 0, 0, -1.419, 47.96, .189; 0, 0, 0, -7.017, .189, 47.23];
function [Nat_Frequency_HZ,ModeShapes] = Nat_Frequency_ModeShape_Calculator(K,M)
% User Input
%This is a subfunction of the main program of PT_Wizard_Calculator.
%Please see the main routine document for further information of inputs.
%  
%   K: Stiffness Matrix of the system [6x6 matrix]
%   M: The Mass System Matrix
% Output:
%   Nat_Frequency: Natural Frequencies of the system [6x1 dimension]
%   ModeShapes: ModeShapes of the System [6x6 dimension]
%Solving Eigenvalue Problem K*x=lambda*M*x 
[EigenVec,EigenVal]=eig(K,M);
%Natural Frequency (rad/sec) are the Eigenvalues
Nat_Frequency_RS=diag(EigenVal);
%Sorting Frequency in ascending order
[Nat_Frequency_RS,Indexing]=sort(sqrt(Nat_Frequency_RS));
%Converting rad/sec to hz
Nat_Frequency_HZ=Nat_Frequency_RS/(2.*pi);
%ModeShapes are the Eigenvectors
ModeShapes=EigenVec(:,Indexing);
ModeShapes=ModeShapes';

The eigenvalues in nastran are: [0.9 -0.1 1.1 0 0 0; -0.1 -1.7 0.1 0 0 0; 1.4 -0.2 -1.2 0 0 0; 0 -0.2 0 0 0 0; 0.8 0.1 0.9 0 0 0; 0 0.5 -0.1 0 0 0]

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Muthu Annamalai il 3 Lug 2013

Seems like a case of differing by a unitary transformation between the eigenvectors from MATLAB ans nastran. I would only say you can apply a Gram-Schmidt orthonormalization procedure to both sets and compare them. This should give you a similar (same?) basis set.

Melissa il 3 Lug 2013