This presentation will discuss the following three topics:

- The Importance of the Eigenvalue Problem in Science and Engineering
- Definition of the Algebraic Eigenvalue Problem and the Need for Numerical/Computational Techniques
- Numerical Solution Techniques

The eigenvalue problem has applications many fields, including

- Quantum Mechanics
- Control Theory
- Solid Mechanics and Stress
- The Study of Vibrations and Resonance

Given and *nxn* matrix A and an *nx1* nonzero vector **x**, the n values of λ for which

are known as the eigenvalues of A. Subtracting λ**x** from both sides of the equation results in

Because **x** is a solution to the above homogeneous system and because **x**≠**0**, the matrix

cannot be invertable and therefore

This last equation is known as the characteristic equation of A and its roots are the eigenvalues of A.

Consider the triangular area shown below.

Given that the moments of Inertia of the area are

The two-dimensional moment of inertia matrix can be written as

so that the eigenvalues are the area's principal momements of inertia. Using det(A-λI_{n})=0, the characteristic equation is

In polynomial form,

The principal moments of inertia are therefore

- Finding the determinants of matrices (even if small) is computationally expensive ( O(3) ).
- If A happens to have n≥5, then det(A - λI
_{n}) will be a polynomial of degree n≥5.

- Advantages
- Easy to understand and code
- Can be used to find one eigenvalue and its (usually the dominant eigenvalue λ
_{1}) associated eigenvector simultaneously - Disadvantages
- Converges slowly unless a good approximation of the eigenvector associated with λ
_{1}is known - λ
_{1}can neither be a repeated root nor a complex root of the characteristic equation

- Advantages
- Generally converges very quickly
- Finds all eigenvalues simultaneously
- Disadvantages
- More challenging to understand and code than power method

Two matrices A and B are similar and have the same eigenvalues if there is an invertible matrix P such that

If a matrix Q is orthogonal, then

(i.e., the transpose of Q is the inverse of Q)

A triangular matrix R can come in two flavors. The first is called lower triangular, in which all entries above the main diagonal are zero. The second is called upper triangular, in which all the entries *below* the main diagonal are zero.

Triangular matricies have a lot of interesting properties. For example, the transpose of a lower triangular matrix is an upper triangular matrix and vice versa. Also, the diaganal entries of a triangular matrix R are the eigenvalues of R.

*Example - Upper Triangular Matrix*

The eigenvalues of R are c_{1}, c_{2}, and c_{3}.

Starting with an *n x n* symmetric matrix A_{1}, the QR factorization technique
begins by factoring or decomposing A_{1} into an orthogonal matrix Q_{1} and an
upper right triangular matrix R_{1} to obtain

Multiplying R_{1} from the right by Q_{1} results in a second matrix.

Substituting gives

More generally,

- A shift quantity s can be introduced to speed up convergence. This method is sometimes called the sigle-shift algorithm.
- Even with shifting, the cost of a QR transformation is still very high using this approach. A better technique is to reduce A to Hessenberg form before decomposing A into Q and R components. The result higher effeciency and faster convergence.

The matrix A has eigenvalues of 0, 1, 2, and 3

Result of 1 iteration:

Result of 2 iterations:

Result of 8 iterations:

After 8 iterations, all but two zeros have emerged completely and one eigenvalue has been computed to full accuracy.

- The eigenvalue problem has applications across a diverse range of quantitative fields
- For very small n the definition may be used as a solution technique. For
*2 < n < 5*the definition is cumbersome and unnecessary. For*n≥5*the definition is at best inadequate. - The QR method is a fast, modern solution technique that can be used find all the eigenvalues of a matrix simultaneously.