Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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How to solve question 3?
![1 The Cayley-Hamilton Theorem gives us a method of finding powers of a matrix. For example, if A is a 2×2
matrix and has characteristic equation A²+bx+c=0 then A²+bA+cI = 0, hence A² = -bA - cI.
(a) Verify this result for the matrix [19].
(b) By multiplying throughout by A we get A³ = -6A² - cA which can be evaluated using the result of
part (a). Continue in this way to calculate A³ and A4.
2 A square matrix A is nilpotent if Ak = 0 for some positive integer k. What can you say about the eigenvalues
of a nilpotent matrix?
3 (a)
Assuming that b ‡ 0, find an orthogonal matrix that diagonalises the matrix A = [%]. That is, find
an orthogonal matrix P such that PT AP is a diagonal matrix.
(b)
=
Determine the sign property of the quadratic form Q: R²R given by Q(x, y)
(i) completing the square,
(ii)
using the eigenvalue test,
(iii) using Sylvester's criteria.
x² + 4xy + y², by
(c)
[X],
Using a linear transformation x = ÎX, where  is an orthogonal matrix, x = [] and X
express Q in diagonal form in the variables X and Y, and then in the original variables x and y.
For the matrix
4 (a)
2 -1
-1 2 -1
-1 -1 2
=
B =
find an orthogonal matrix P such that PT BP is a diagonal matrix.
(b) Write down the sign property of ¢ = 2x² + 2y² + 2z² - 2xy - 2yz - 2zx. As in 3(c), use an orthogonal
transformation to express o in diagonal form. What does the equation p(x, y, z) = 0 mean?](https://content.bartleby.com/qna-images/question/dc3c0612-e198-45c8-8e80-98d06c0b64b4/83edaf8a-fdbf-4810-9c8d-30517aa5cedc/3y16g4e_processed.png)
Transcribed Image Text:1 The Cayley-Hamilton Theorem gives us a method of finding powers of a matrix. For example, if A is a 2×2
matrix and has characteristic equation A²+bx+c=0 then A²+bA+cI = 0, hence A² = -bA - cI.
(a) Verify this result for the matrix [19].
(b) By multiplying throughout by A we get A³ = -6A² - cA which can be evaluated using the result of
part (a). Continue in this way to calculate A³ and A4.
2 A square matrix A is nilpotent if Ak = 0 for some positive integer k. What can you say about the eigenvalues
of a nilpotent matrix?
3 (a)
Assuming that b ‡ 0, find an orthogonal matrix that diagonalises the matrix A = [%]. That is, find
an orthogonal matrix P such that PT AP is a diagonal matrix.
(b)
=
Determine the sign property of the quadratic form Q: R²R given by Q(x, y)
(i) completing the square,
(ii)
using the eigenvalue test,
(iii) using Sylvester's criteria.
x² + 4xy + y², by
(c)
[X],
Using a linear transformation x = ÎX, where  is an orthogonal matrix, x = [] and X
express Q in diagonal form in the variables X and Y, and then in the original variables x and y.
For the matrix
4 (a)
2 -1
-1 2 -1
-1 -1 2
=
B =
find an orthogonal matrix P such that PT BP is a diagonal matrix.
(b) Write down the sign property of ¢ = 2x² + 2y² + 2z² - 2xy - 2yz - 2zx. As in 3(c), use an orthogonal
transformation to express o in diagonal form. What does the equation p(x, y, z) = 0 mean?
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