Available online
Description
This programme shows that any symmetric n x n matrix can be diagonalised. (This programme was re-edited in 1995 as FOUM473A)
This programme shows that any symmetric n x n matrix can be diagonalised. (This programme was re-edited in 1995 as FOUM473A)
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 12 |
| First transmission date: | 08-05-1980 |
| Published: | 1980 |
| Rights Statement: | Rights owned or controlled by The Open University |
| Restrictions on use: | This material can be used in accordance with The Open University conditions of use. A link to the conditions can be found at the bottom of all OUDA web pages. |
| Duration: | 00:21:52 |
| + Show more... | |
| Producer: | David Saunders |
| Contributors: | Norman Gowar; Robin Wilson |
| Publisher: | BBC Open University |
| Keyword(s): | Determinant; Group; Matrix |
| Subject terms: | Matrices; Symmetric matrices |
| Footage description: | Norman Gowar discusses how matrices could be simplified to make calculations more simple. As every matrix represents a linear transformation it may be possible to choose a different basis for the transformation to get a simpler matrix. A diagonal matrix is a particularly easy matrix for calculations. The aim of the programme is to see if a symmetric matrix always allows us to diagonalise the matrix. Robin Wilson tests to see if all 2 by 2 symmetric matrices can be diagonalised. Using animations he demonstrates what is happening as the matrix is diagonalised. Whenever there is a diagonal matrix then each basis vector is mapped to a multiple of itself. Such a vector is called an eigenvector. Using animations he shows how vectors are represented as eigenvectors. He then describes the corresponding change in the matrix formation and argues that for any two symmetric matrices it is possible to find two eigen values and hence diagonalise the matrix. He now works through a proof of why this is so. The eigen vector equation always resolves itself as a quadratic equation giving two roots. Using animations he shows graphically why this is so. Norman Gowar relates this work to changes in basis and transition matrices. He shows that the eigen vectors of a symmetric matrix are orthogonal and that the transition matrix mapping two matrices A and B to each other has columns which are orthonormal. Such a matrix is called an orthogonal matrix. Where two matrices A and B are related by an orthogonal matrix, then if A is symmetric, B also is symmetric. Robin Wilson now works through an example to show that a symmetric 3 by 3 matrix can be diagonalised. The eigenvector equation always resolves itself as a cubic equation, giving three real roots. Using animations he demonstrates this graphically, showing that there must always be at least one eigenvector as the basis vector, this in turn allows the basis to be rotated, giving the other eigenvectors and showing that the matrix can be diagonalised, by comparison with the 2 by 2 matrix example. Norman Gowar generalises the results of the programme arguing that any n x n symmetric matrix can be diagonalised. |
| Master spool number: | HOU2934 |
| Production number: | FOU9081N |
| Available to public: | no |
