{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lect3-06-web

# Lect3-06-web - MATH 304 Fall 2011 Linear Algebra Homework...

This preview shows pages 1–10. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #10 (due Thursday, November 17) All problems are from Leon’s book (8th edition). Section 5.5: 6c, 29a, 29b, 30a, 30b Section 5.6: 1a, 3, 4, 7, 8 MATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial. Eigenvalues and eigenvectors of a matrix Definition. Let A be an n × n matrix. A number λ ∈ R is called an eigenvalue of the matrix A if A v = λ v for a nonzero column vector v ∈ R n . The vector v is called an eigenvector of A belonging to (or associated with) the eigenvalue λ . Remarks. • Alternative notation: eigenvalue = characteristic value , eigenvector = characteristic vector . • The zero vector is never considered an eigenvector. Diagonal matrices Let A be an n × n matrix. Then A is diagonal if and only if vectors e 1 , e 2 , . . . , e n of the standard basis for R n are eigenvectors of A . If this is the case, then the diagonal entries of the matrix A are the corresponding eigenvalues: A = λ 1 O λ 2 . . . O λ n ⇐⇒ A e i = λ i e i Eigenspaces Let A be an n × n matrix. Let v be an eigenvector of A belonging to an eigenvalue λ . Then A v = λ v = ⇒ A v = ( λ I ) v = ⇒ ( A − λ I ) v = . Hence v ∈ N ( A − λ I ), the nullspace of the matrix A − λ I . Conversely, if x ∈ N ( A − λ I ) then A x = λ x . Thus the eigenvectors of A belonging to the eigenvalue λ are nonzero vectors from N ( A − λ I ). Definition. If N ( A − λ I ) negationslash = { } then it is called the eigenspace of the matrix A corresponding to the eigenvalue λ . How to find eigenvalues and eigenvectors? Theorem Given a square matrix A and a scalar λ , the following statements are equivalent: • λ is an eigenvalue of A , • N ( A − λ I ) negationslash = { } , • the matrix A − λ I is singular, • det( A − λ I ) = 0. Definition. det( A − λ I ) = 0 is called the characteristic equation of the matrix A . Eigenvalues λ of A are roots of the characteristic equation. Associated eigenvectors of A are nonzero solutions of the equation ( A − λ I ) x = . Example. A = parenleftbigg a b c d parenrightbigg . det( A − λ I ) = vextendsingle vextendsingle vextendsingle vextendsingle a − λ b c d − λ vextendsingle vextendsingle vextendsingle vextendsingle = ( a − λ )( d − λ ) − bc = λ 2 − ( a + d ) λ + ( ad − bc ). Example. A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 25

Lect3-06-web - MATH 304 Fall 2011 Linear Algebra Homework...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online