Lecture #2 Notes

# Here multilinearity is key in getting the

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Unformatted text preview: = AB , where A, B ∈ Mn , then det(C ) = det(A)det(B ) An instance where this becomes useful is where C could be a diﬃcult matrix, whereas the factors A or B might be triangular, unitary, etc. Here, multilinearity is key in getting the factorization property. 2 • If A = ￿ ￿ BC , where B ∈ Mm , D ∈ Mn−m , then 0D det(A) = det(B )det(D) We also have det(A) = n ￿ j =1 (−1)i+j ai,j det(Ai,j ) where the matrix Ai,j is obtained by deleting the ith row and j th column from matrix A. • By AA−1 = I , we have det(AA−1 ) = det(I ) = 1 Since det(AA−1 ) = det(A)det(A−1 ), we have that A is invertible ⇒ det(A) ￿= 0. We will see further below that the converse is true as well. 1.3 Eigenvales and Eigenvectors 1.3.12 Deﬁnition. If A ∈ Mn and ∃λ ∈ Cm and x ∈ Cn \{0} such that Ax = λx, then, λ is an eigenvalue of A and x is the corresponding eigenvector. Given any such λ, the set {y : Ay = λy } is called the eigenspace corresponding to λ and the set of eigenvalues i...
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## This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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