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Presentation on Eigenvalue Problem

Discussion Outline

This presentation will discuss the following three topics:

  1. The Importance of the Eigenvalue Problem in Science and Engineering
  2. Definition of the Algebraic Eigenvalue Problem and the Need for Numerical/Computational Techniques
  3. Numerical Solution Techniques

Importance - Why solve the eigenvalue problem?

The eigenvalue problem has applications many fields, including


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 λx from both sides of the equation results in


Because x is a solution to the above homogeneous system and because x0, the matrix

cannot be invertable and therefore

Characterisic Equation of A

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-λIn)=0, the characteristic equation is


In polynomial form,


The principal moments of inertia are therefore



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

Numerical Techniques

The Power Method

QR Method

Special Matrices

Similar Matricies

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

Orthogonal Matrices

If a matrix Q is orthogonal, then

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

Triangular Matrices

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 c1, c2, and c3.

QR Algorithm

Starting with an n x n symmetric matrix A1, the QR factorization technique begins by factoring or decomposing A1 into an orthogonal matrix Q1 and an upper right triangular matrix R1 to obtain

Multiplying R1 from the right by Q1 results in a second matrix.

Substituting gives

More generally,



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.