Module 18 - Matrix Analysis 3 (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 18

MATRICES III

Module Topics 1. Rank 2. Eigenvalues and eigenvectors

A:

Work Scheme based on JAMES (THIRD EDITION)

1. Sets of n simultaneous linear equations in n unknowns (in the form AX = b, with b either zero or non-zero) were considered in Module 17. When |A| 6= 0 it proved easy to obtain solutions. However, when |A| = 0 it was much harder to assess the position, and to help clarify the situation it is necessary to define the rank of a matrix. Turn to p.329 and study section 5.6, stopping before the final paragraph on p.330. As indicated at the top of p.329 the first step in simplifying the system (5.33) is to consider the first column and make zero all elements below a11 . Thus, you evaluate (row 3 – row 1), (row 4 – row 1) and (row 6 – row 1). You then proceed in a similar way for the second and subsequent columns. J. states on p.330 that for a non-singular matrix the elimination method leads to an upper triangular matrix, with non-zero diagonal elements. For a singular matrix, however, such a reduction is not possible and the best you can do is to simplify to a matrix in echelon form, similar to that shown in the middle of p.330. The number of non-zero rows in the echelon form of a matrix A is the rank of the matrix. The rank of A gives the number of essentially different rows in the matrix (i.e. rows which cannot be written as linear combinations of some or all of the other rows). Understanding this point is extremely important in determining the types of solution that are possible for general sets of equations of the form AX = b. As stated near the bottom of p.330, the mathematical definition of rank involves non-zero determinants of submatrices. For a large matrix such calculations can be difficult to carry out and the reduction to echelon form is usually a better way of calculating the rank of a matrix. 2. Continue studying section 5.6, on the consistency or otherwise of a set of equations, from the fourth line of p.331 to just above Example 5.35. The final result is the shaded paragraph on p.332. Work through Example 5.35. Note that in using the elimination method J. also adopts partial pivoting, which is discussed at the bottom of p.317. When dealing with a particular column of a non-singular matrix it is found to be more accurate to rearrange the equations (or rows) first to ensure that the element with the largest magnitude lies on the main diagonal. Then divide that equation (or row) by the value of the element to make the element 1 (i.e. make the pivot unity). Then proceed with the standard elimination method for that column. Move on to the next column and again first rearrange the rows as above, if necessary. Partial pivoting can also be used, in an analogous way, when reducing singular matrices to echelon form. J. uses partial pivoting in both reductions in Example 5.35, although it is not necessary to do so to obtain the stated answers. 3. Study the remainder of section 5.6, starting below the horizontal line on p.333. This work looks at simple cases of m equations in n unknowns and shows how to decide whether a solution exists and, if it –1–

does, how to determine it. note that a set of equations is underspecified if there are more variables than equations, and overspecified if there are more equations than variables. ***Do Exercises 64, 66(a),(b) on p.336*** 4. The second half of this module is on the calculation of eigenvalues and eigenvectors. Turn to p.337 and study the introduction to section 5.7. Next study section 5.7.1, noting that in this module the characteristic equation will usually be written in the form |A − λI| = 0 . Work through Examples 5.36 and 5.37, and read the final sentence on p.339. 5. Study section 5.7.2 on eigenvalues and eigenvectors, and then work through Example 5.38.     1 j respectively, and B Study Example 5.39, noting that the eigenvectors should be written A j 1 where A and B are constants. Study Example 5.40. In obtaining the eigenvector e1 the components of the vector satisfy the 3 equations stated near the bottom of p.342. By choosing e 13 = β1 it then follows from these equations that e12 = 3 β1 1 and e11 = β1 . Hence the solution is e1 = β1  3 , where β1 is a constant. Similar analyses hold when 1 determining the other eigenvectors. 6. Read the comments below the horizontal line on p.344 on normalising the eigenvectors. 7. Before finishing the module two further points are briefly mentioned. When A is symmetric, a situation which occurs in many applications, then the eigenvalues are real and the eigenvectors corresponding to different eigenvalues are perpendicular as stated in section 5.7.7 on p.351. It should also be observed that in all the cases considered in this module the roots of the characteristic equation are real and distinct. Clearly this is not always the case, and when the roots are equal the calculation of the eigenvectors is slightly less straightforward. Time is not available to investigate equal (or repeated) roots in this module. ***Do Exercise A:  (i)

B:

5 0

1 4



Find the eigenvalues and eigenvectors of the matrices     1 0 0 −3 1 −1 (ii)  0 3 0  (iii)  1 −5 1 0 2 4 −1 1 −3

Work Scheme based on STROUD (FIFTH EDITION)

The first part of this module, on calculating the rank of a matrix and finding solutions to systems of m equations in n unknowns, is not in S. so you must work through sections 1–3 of A: Work Scheme based on JAMES (THIRD EDITION). Next turn to p.558 of S. and work through frames 47–54 on eigenvalues and eigenvectors. S. mentions in the text that an eigenvector can always be multiplied by a constant but he then assumes it to be unity in writing down his solutions. You are expected to include the non-zero constant in your answers (see the worked solutions in the module). –2–

Specimen Test 18  1.

2.

Find the rank of the matrix

1  2  −1 2

1 3 1 3

−1 −3 0 −2

 1 3 . 0 2

Solve, where possible, the following matrix equations:    x    3 2 1 1   y = (i) 5 4 2 2 z  (ii)

   2 1   3 1 1 x = 1 y 3 4 2 

3.

Find the eigenvalues and eigenvectors of the matrix

–3–

3 0 1

 0 1 3 1 . 1 2