1 / 48

Lecture 5

Lecture 5. Today Evaluation of a Determinant using E. O. Properties of Determinants Introduction to Eigenvalues Applications Reading Assignment : Secs 3.3 - 3.5 of Textbook Homework #3 assigned. Elementary Linear Algebra

lail
Download Presentation

Lecture 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 5 Today Evaluation of a Determinant using E. O. Properties of Determinants Introduction to Eigenvalues Applications Reading Assignment: Secs 3.3 - 3.5 of Textbook Homework #3 assigned Elementary Linear Algebra Larsen & Falvo (6th Edition) TKUEE翁慶昌-NTUEE SCC_10_2008

  2. Lecture 5 Next Time Vectors in Rn Vector Spaces Subspaces of Vector Spaces Reading Assignment: Secs 4.1 - 4.3 of Textbook Elementary Linear Algebra Larsen & Falvo (6th Edition) TKUEE翁慶昌-NTUEE SCC_10_2008

  3. What have you learned about E.O. and Determinants? • Q1: Why can every invertible matrix be represented by mutiplications of elementary matrices? • Q2: How is determinant related to systems of linear equations?

  4. Geometry of Determinants: Determinants as Size Functions • We have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of that determinant. One way to compute the area that it encloses is to draw this rectangle and subtract the area of each subregion.

  5. The region formed by and is bigger, by a factor of k, than the shaded region enclosed by and . That is, size ( , ) = k ·size( , ) and in general we expect of the size measure that size(. . . , , . . . ) = k ·size(. . . , , . . . ). • The properties in the definition of determinants make reasonable postulates for a function that measures the size of the region enclosed by the vectors in the matrix. See this case:

  6. Another property of determinants is that they are unaffected by pivoting. Here are before-pivoting and after-pivoting boxes (the scalar used is • k = 0.35). Although the region on the right, the box formed by and , is more slanted than the shaded region, the two have the same base and the same height and hence the same area. This illustrates that

  7. That is, we’ve got an intuitive justification to interpret det ( , . . . , ) as the size of the box formed by the vectors. Example The volume of this parallelepiped, which can be found by the usual formula from high school geometry, is 12.

  8. The only difference between them is in the order in which the vectors are taken. If we take first and then go to , follow the counterclockwise are shown, then the sign is positive. Following a clockwise are gives a negative sign. The sign returned by the size function reflects the ‘orientation’ or ‘sense’ of the box.

  9. Volume, because it is an absolute value, does not depend on the order in which the vectors are given. The volume of the parallelepiped in the following example, can also be computed as the absolute value of this determinant. The definition of volume gives a geometric interpretation to something in the space, boxes made from vectors.

  10. 3.2 Evaluation of a determinant using elementary operations • Thm 3.3: (Elementary row operations and determinants) Let A and B be square matrices.

  11. Ex:

  12. Sol: Note: A row-echelon form of a square matrix is always upper triangular. • Ex 2: (Evaluation a determinant using elementary row operations)

  13. Notes:

  14. Notes:

  15. Thm 3.4: (Conditions that yield a zero determinant) If A is a square matrix and any of the following conditions is true, then det (A) = 0. (a) An entire row (or an entire column) consists of zeros. (b) Two rows (or two columns) are equal. (c) One row (or column) is a multiple of another row (or column).

  16. Ex:

  17. Note: Number of operations for cofactor expansion of nxn matrix ~ n! 30! = ?

  18. Ex 5: (Evaluating a determinant) Sol:

  19. Ex 6: (Evaluating a determinant) Sol:

  20. 3.3 Properties of Determinants • Notes: • Thm 3.5: (Determinant of a matrix product) det (AB) = det (A) det (B) (1) det (EA) = det (E) det (A) (2) (3)

  21. Ex 1: (The determinant of a matrix product) Find |A|, |B|, and |AB| Sol:

  22. Check: |AB| = |A| |B|

  23. Ex 2: • Thm 3.6: (Determinant of a scalar multiple of a matrix) If A is an n × n matrix and c is a scalar, then det (cA) = cn det (A) Find |A|. Sol:

  24. Thm 3.7: (Determinant of an invertible matrix) • Ex 3: (Classifying square matrices as singular or nonsingular) A square matrix A is invertible (nonsingular) if and only if det (A)  0 Sol: A has no inverse (it is singular). B has inverse (it is nonsingular).

  25. Ex 4: • Thm 3.8: (Determinant of an inverse matrix) • Thm 3.9: (Determinant of a transpose) (a) (b) Sol:

  26. If A is an n × n matrix, then the following statements are equivalent. • Equivalent conditions for a nonsingular matrix: (1) A is invertible. (2) Ax = b has a unique solution for every n × 1 matrix b. (3) Ax = 0 has only the trivial solution. (4) A is row-equivalent to In (5) A can be written as the product of elementary matrices. (6) det (A)  0

  27. Ex 5: Which of the following system has a unique solution? (a) (b)

  28. Sol: (a) This system does not have a unique solution. (b) This system has a unique solution.

  29. Eigenvalue and eigenvector: A：an nn matrix ：a scalar x： a n1nonzero column matrix Eigenvalue Eigenvector (The fundamental equation for the eigenvalue problem) 3.4 Introduction to Eigenvalues • Eigenvalue problem: If A is an nn matrix, do there exist n1 nonzero matrices x such that Ax is a scalar multiple of x？

  30. Eigenvalue Eigenvalue Eigenvector Eigenvector • Ex 1: (Verifying eigenvalues and eigenvectors)

  31. Note: (homogeneous system) If has nonzero solutions iff . • Characteristic equation of AMnn: • Question: Given an nn matrix A, how can you find the eigenvalues and corresponding eigenvectors?

  32. Sol: Characteristic equation: Eigenvalue: • Ex 2: (Finding eigenvalues and eigenvectors)

  33. Ex 3: (Finding eigenvalues and eigenvectors) Sol: Characteristic equation:

  34. 3.5 Applications of Determinants • Matrix of cofactors of A: • Adjoint matrix of A:

  35. Thm 3.10: (The inverse of a matrix given by its adjoint) If A is an n × n invertible matrix, then • Ex:

  36. Ex 2: (a) Find the adjoint of A. (b) Use the adjoint of A to find Sol:

  37. adjoint matrix of A inverse matrix of A • Check: cofactor matrix of A

  38. Thm 3.11: (Cramer’s Rule) (this system has a unique solution)

  39. ( i.e. )

  40. Pf: A x = b,

  41. Ex 6: Use Cramer’s rule to solve the system of linear equations. Sol:

  42. Keywords in Section 3.5: • matrix of cofactors : 餘因子矩陣 • adjoint matrix : 伴隨矩陣 • Cramer’s rule : Cramer 法則

  43. Today • Inverse of a Matrix • Elementary Matrices • Determinant of a Matrix • Evaluation of a Determinant using Elementary Operations Reading Assignment: Secs 2.5 & 3.1-3.2 of Textbook Homework #2 Due and #3 Assigned Next Time (10/24, Class 3:30pm – 6:20pm) • Evaluation of a Determinant using E. O. (Cont.) • Properties of Determinants • Introduction to Eigenvalues • Applications Reading Assignment: Secs 3.2 - 3.5 of Textbook

More Related