Application of eigen value eigen vector to design

An eigenvalue of a square matrix is a scalar that is usually represented by the Greek letter λ (pronounced lambda).

1. Application of Eigen value Eigen Vector to Design Bridge Presented by Talha momin (33) Sugam pandey (34) Atharva parab (35) Simran pardeshi (36)

2. AGENDA Introduction Definition of Eigen value & Eigen vector Application for Designing Bridge Other Applications Conclusion

3. INTRODUCTION In the 18 th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Later Joseph-Louis Lagrange realized that the principal axes are eigenvectors of the inertia matrix. In 19 th century Cauchy used there work to classify quadric surface and name this term as characteristic root and now they are called as eigenvalues. Eigen is the German word meaning ‘proper’ or characteristics.

4. DEFINITION An eigenvalue of a square matrix is a scalar that is usually represented by the Greek letter λ (pronounced lambda). As you might suspect, an eigenvector is a vector. Moreover, we require that an eigenvector be a non-zero vector, in other words, an eigenvector can not be the zero vector. We will denote an eigenvector by the small letter x. All eigenvalues and eigenvectors satisfy the equation for a given square matrix

5. Linear algebra studies linear transformation ,which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and Eigen spaces are properties of a matrix. In general, a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged or possibly reversing it. These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with that Eigenvector.

6. Application for designing bridge The natural frequency of the bridge is described by the eigenvalue of smallest magnitude. The engineers exploit this knowledge to ensure the stability of their constructions. Equation for natural frequencies that must be solved: [M]{ü}+[K]{u} = 0 where, [M] = mass matrix [K] = stiffness matrix {ü} = 2nd derivative of {u} {u} = {ø}sin ω t {ø} = eigenvector ω = (λ) circular natural frequency ^^ need to solve for ω

7. USING EIGENVALUES AND EIGENVECTORS ENGINEERS CAN PREVENT DISASTERS.THOUGHTACOMA BRIDGE 1940 COULDN'T BESAVED.

8. TACOMA BRIDGE 1940 EIGEN VALUES AND EIGEN VECTORS ARE USED FOR DESIGNING BRIDGES. TACOMA BRIDGE BUILD IN 1940 WAS ONE OF THE EXAMPLES OF DESIGNING BRIDGE USING EIGENVALUES TO MAKE BRIDGE MORE DURABLE. HOWEVER,AN INCIDENT TOOK PLACE IN 7 NOVEMBER 1940 ON TACOMA BRIDGE WHERE THEWIND OF (42 MPH) HIT THE BRIDGE AND UNFORTUNATELY THE NATURAL FREQUENCY OF BRIDGE MET THE FRRQUENCY OF WIND AND DUE TO WHICH RESONANCE WAS CREATED BETWEEN THEM AND THE BRIDGE COLLAPSED, JUST 4 MONTHS AFTER IT WAS BUILT. TACOMA BRIDGE IS SAID TO BE AN FAILURE IN ENGINEERS CREATIONS, AS NO PROPER IMPORTANCE WAS GIVEN TO AERODYNAMICS OF BRIDGE.

9. THE CRASH

10. Other Applications

11. Eigenvalue analysis is also used in the design of the car stereo systems, where it helps to reproduce the vibration of the car due to the music. Meridian sound system which is mostly used in luxury cars are simply based on the principle of eigen values and eigen vectors DESIGNING CAR STEREO SYSTEM :

12. Mechanical Engineering : Eigenvalues and eigenvectors allow us to "reduce“ a Linear operation to separate, simpler, problems. For example, If a stress is applied to a "plastic" solid, the deformation can be dissected in to "principle directions"-those directions in which the deformation is greatest. Vectors in the principle directions are the Eigenvectors and the percentage deformation in each principle

13. Oil Companies Oil companies frequently use eigenvalue analysis to explore land for oil. Oil , dirt, and other substances all give rise to linear systems which have different eigenvalues, so eigenvalue analysis can give a good indication of where oil reserves are located.  Oil companies place probes around a site to pick up the waves that result from a huge truck used to vibrate the ground.  The waves are changed as they pass through the different substances in the ground. The analysis of these waves directs the oil companies to possible drilling sites.

14. FUTHER APPLICATIONS Vibration analysis Civil Engineering Face recognition Musical instruments

15. CONCLUSION Eigenvalues and Eigenvectors are fundamental in data science and model-building in general. Besides their use in PCA, they are employed, namely, in spectral clustering and image compression. Hence, it is important to have clear in mind their geometrical interpretation.

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