MATH 3A Study Guide - Final Guide: Diagonal Matrix, Diagonalizable Matrix, Vehicle-To-Grid

• School: UC-Irvine
• Department: Mathematics
• Course: MATH 3A
• Professor: All

Document Summary

1. 1: (a) we have det ( i3 a) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1 2 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 2 1. = (cid:0) 2 2 + 1(cid:1) ( + 1) + (1 ) = ( 1)2 + ( 1) ( 1) = ( 1)2, so the characteristic equation is ( 1)2 = 0. (b) we will nd a basis for each eigenspace. A 0i3 = which has reduced echelon form . Moving on to the 1-eigenspace, it"s the null space of is a basis for the 0-eigenspace. A 1i3 = which has reduced echelon form (cid:110)(cid:104) 1 1. Since only the rst column is pivot, this eigenspace has dimension 2 and can be parametrized by the x2 and x3 coordinates. The coordinates of an element of the eigenspace must satisfy x1 + x2 + x3 = 0, so we nd x2 x3 (cid:104) x1 (cid:105) Finally, we know without any further calculation that the set is a basis for the 1-eigenspace.