Real eigenvalues and eigenvectors | Lecture 33 | Matrix Algebra for Engineers

Finding distinct real eigenvalues of a 2x2 matrix.

[kyuubiegi]

Genuinely one of the cleanest, easiest to understand example i've ever seen. Thanks, keep it up!

[Selinzik_CEO]

This guy deserves a Noble Prize in Solving Problems

[CorbettMatthews]

Jeff, I might actually pass my linear midterm tomorrow. Thanks for your help!

[adakoleowoicho]

Thank you sir, u have made it very simple and clear for me

[roger]

Suppose that I have a 1 cm thick by 2 cm wide by 10 cm long solid aluminum bar with the origin of a Cartesian coordinate system located at its center of mass and oriented randomly relative to the symmetric centerlines of the bar. Then I will get some 3 by 3, symmetric, positive definite inertia tensor, and the three positive eigenvalues associated with this matrix will equal the three principal moments of inertia of this bar. The three eigenvectors, normalized to unity, may be assembled into three columns of a 3 by 3 direction cosine matrix that will transform the original arbitrary inertia tensor into a diagonal inertia tensor. Axes of the related Cartesian coordinate system will be aligned with the bar's lines of symmetry, but an arbitrary arrangement of eigenvectors and even rescaling any of them by –1 will achieve a range of feasible coordinate axes arrangements, and even give transformation matrices with negative determinants. It looks like one must have in mind the final desired coordinate system orientation and rearrange the order and scaling of the eigenvectors to achieve the final desired direction cosine matrix. Is there any other guidance on scaling and rearranging eigenvectors to get the desired direction cosine matrix?

[mouctechy]

you're really a good lecturer thanks so much
for the clarification

[onesimus]

Thanks a lot, I appreciate your teaching

[roger]

Assembling the eigenvectors normalized to unity into a square matrix forms an orthonormal matrix that transforms the original matrix into a diagonal matrix containing the eigenvalues into the user assigned order of eigenvalues and eigenvectors. The determinant of this matrix may be ±1, depending on the eigenvector arrangement. Is there any way to associate the individual eigenvalues with the respective rows and columns of the symmetric matrix so the transformation matrix determinant will always be +1?