Determinants and eigenvalues Math 40, Introduction to Linear Algebra Wednesday, February 15, 2012 Amazing facts about determinants det A can be found by “expanding” along any row or any column Theorem. The determinant of a triangular Consequence: matrix is the product of its diagonal entries. 123 4 056 7 A = det(A)=1 5 8 10 = 400 008 9 · · · 0 0 0 10 Amazing facts about determinants EROs barely change the determinant, and they do so in a predictable way. EROs effect on det A Strategy to compute swap two rows changes sign det A more quickly for general matrices A = multiply row multiply det ⇒ by scalar c by scalar c Perform EROs to get REF of A and add c ຘ row i no change at all! compute det A based to row j on det of REF Amazing facts about determinants Theorem. A square matrix A is invertible if and only if det A =0. det(A) = det(AT ) determinant is multiplicative det(AB) = det(A) det(B) Consequence: 1 1 det(A) det(A− ) = det(AA− ) = det(I)=1 det(A 1)= 1 ⇒ − det A 1 1 If A is invertible, det(A− )= det A . Example using properties of determinant Example If det A = -3 for a 5 x 5 matrix A, find the determinant of the matrix 4A3. We have det(4A3) =45 det(A3) =45[det(A)]3 = 45( 3)3 − Another property of the determinant? Question True or false: det(A + B) = det A + det B ? False! Consider 11 1 1 A = ,B= − 11 11 − Then 20 A + B = det(A + B)=4= 0 = det A + det B 02 Eigenvalues and eigenvectors Introduction to eigenvalues Let A be an n x n matrix. If Ax = λx for some scalar λ and some nonzero vector x, then we say λ is an eigenvalue of A and x is an eigenvector associated with λ. n n Viewed as a linear transformation from R to R A sends vector x to a scalar multiple of itself ( λ x ) . Eigenvalues, eigenvectors for a 2x2 matrix 12 A = 54 12 2 12 2 Notice that = =6 54 5 30 5 12 6 6 =6 54 15 15 eigenvectors eigenvalues 6 2 =3 12 1 1 1 15 5 − = =( 1) − 54 1 1 − 1 − Any (nonzero) scalar multiple of an eigenvector is itself an eigenvector (associated w/same eigenvalue). A(cx)=c(Ax) = c(λx)=λ(cx) Graphic demonstration of eigenvalues and eigenvectors: eigshow eigshow demonstrates how the image A x changes as we rotate a unit vector x in R 2 around a circle in particular, we are interested in knowing when A x is parallel to x Finding eigenvalues of A We want nontrivial solutions to Ax = λx Ax λx = 0 Ax λIx = 0 ⇐⇒ − ⇐⇒ − (A λI)x = 0 When does this ⇐⇒ − homogeneous system have a solution other than x = 0 ? Must have that A λ I is not invertible, − which means that det(A λI)=0 − eigenvalues of A given eigenvalue λ, ⇒ associated eigenvectors are find values of λ nonzero vectors in such that det(A λI)=0 null(A λI) − − Example of finding eigenvalues and eigenvectors Find eigenvalues and 10 1 Example − A = 2 15 corresponding eigenvectors of A. − 00 2 1 λ 0 1 − − 0 = det(A λI) = 2 1 λ 5 − − − 002λ − 1 λ 0 characteristic = (2 λ) − − 2 1 λ polynomial − − λ3 +2λ2 + λ 2 − − = (2 λ)(1 λ)( 1 λ) − − − − λ =2, 1, or 1 − Example of finding eigenvalues and eigenvectors Example Find eigenvalues and 10 1 − A = 2 15 corresponding eigenvectors of A. − 00 2 λ =2, 1, or 1 − λ =2 Solve (A 2I)x = 0. − 10 1 0 10 1 0 − − EROs [ A 2I 0]= 2 350 01 1 0 − | − −−−→ − 0000 00 0 0 x x 1 eigenvectors of A for λ = 2 are 1 − 3 − x = x = x3 = x3 1 1 2 − x3 x3 1 c 1 for c =0 for any x3 R 1 ∈ Example of finding eigenvalues and eigenvectors Find eigenvalues and 10 1 Example − A = 2 15 corresponding eigenvectors of A. − 00 2 λ =2, 1, or 1 − 1 − eigenvectors of A for λ = 2 are c 1 for c =0 λ =2 Solve (A 2I)x = 0. − 1 E = eigenspace of A = set of all eigenvectors 0 2 for λ =2 of A for λ =2 ∪ { } = null( A 2I) − 1 − = span 1 1 Example of finding eigenvalues and eigenvectors Example Find eigenvalues and 10 1 − A = 2 15 corresponding eigenvectors of A. − 00 2 λ =2, 1, or 1 − 1 − λ =2 E2 = span 1 1 1 λ =1 Solve (A I)x = 0. = E1 = span 1 − ⇒ 0 0 λ = 1 Solve (A + I)x = 0. = E 1 = span 1 − − ⇒ 0 Eigenvalues, eigenvalues... where are you? 11 Example Find eigenvalues of A. A = 11 − 1 λ 1 0 = det(A λI) = − = (1 λ)2 +1 − 11λ − − − 2 = λ 2λ +2 − 2 √4 8 = λ = ± − =1 i ⇒ 2 ± Eigenvalues are complex!.