MATH125 Lecture Notes - Lecture 31: Gaussian Elimination, Invertible Matrix, Row And Column Spaces

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Nikita

Document Summary

This is the eigenspace e corresponding to . All its nonzero vectors are eigenvectors corresponding to . (4) find a basis for each eigenspace. Find the eigenvalues and the corresponding eigenspaces of. We rst nd the characteristic polynomial: det(a i3) = det . 5 4 ] det[ 0. = ( 2 4 + 5) ( 2) = 3 + 4 2 5 + 2. Next we have to solve equation 3 + 4 2 5 + 2 = 0. One can see that the left hand side is decomposed as ( 1)2( 2). Hence the eigenvalues are = 1 and = 2. To nd the eigenspace corresponding to = 1 we must nd the null space of the matrix. Thus, all eigenvectors corresponding to = 1 are multiple of the vector. The case of = 2 can be considered similarly. Here is the answer: the eigenspace e2 consists of vectors multiple to. Let a and b be n n matrices.