What is special about eigenvalues?

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 into “principle directions”- those directions in which the deformation is greatest.

Are eigen values unique?

Eigenvectors are NOT unique, for a variety of reasons. Change the sign, and an eigenvector is still an eigenvector for the same eigenvalue. In fact, multiply by any constant, and an eigenvector is still that. Different tools can sometimes choose different normalizations.

What are applications of eigenvalues?

Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration analysis, electric circuits, advanced dynamics and quantum mechanics are just a few of the application areas.

Why are eigenvalues important in physics?

The eigenvalues, also important, are called moments of inertia. The eigen functions represent stationary states of the system i.e. the system can achieve that state under certain conditions and eigenvalues represent the value of that property of the system in that stationary state.

Do eigenvalues have to be distinct?

A matrix does not necessarily have distinct eigenvalues (although almost all do), and a matrix does not necessarily have a single eigenvalue with multipicity n. In fact, given any set of n values, you can construct a matrix with those values as eigenvalues (indeed just take the corresponding diagonal matrix).

Are eigen vectors unique?

This is a result of the mathematical fact that eigenvectors are not unique: any multiple of an eigenvector is also an eigenvector! Different numerical algorithms can produce different eigenvectors, and this is compounded by the fact that you can standardize and order the eigenvectors in several ways.

Can eigenvalues be zero?

Eigenvalues may be equal to zero. We do not consider the zero vector to be an eigenvector: since A 0 = 0 = λ 0 for every scalar λ , the associated eigenvalue would be undefined.

Who invented eigenvalues?

Finding the eigenvectors and eigenvalues for a linear transformation is often done using matrix algebra, first developed in the mid-19th century by the English mathematician Arthur Cayley. His work formed the foundation for modern linear algebra.

Why Nxn matrix has n eigenvalues?

All N X N square matrices have N eigenvalues; that’s just the same as saying that an Nth order polynomial has N roots. While a defective matrix still has N eigenvalues, it does not have N independent eigenvectors.