MATH-M 303 Lecture Notes - Lecture 23: If And Only If, Diagonal Matrix, Invertible Matrix
Document Summary
12-7-16 (cid:882) (cid:885): commute in multiplication properties of diagonal matrices (cid:882) (cid:885)(cid:2871)] Recall diagonal matrix- only (possible) nonzero entries are on diagonal. Not all matrices can be diagonal, but class of diagonalizable matrices exists that has many of same: ex. Simple to compute powers: simple, but not as much as for diagonal matrices (cid:882) (cid:885)(cid:2870)] (cid:885)(cid:2870)][(cid:887) (cid:882)(cid:882) (cid:885)]=[(cid:887)(cid:2871) (cid:882) (cid:882) [(cid:887) (cid:882)(cid:882) (cid:885)](cid:2870)=[(cid:887) (cid:882)(cid:882) (cid:885)][(cid:887) (cid:882)(cid:882) (cid:885)]=[(cid:887)(cid:2870) (cid:882) [(cid:887) (cid:882)(cid:882) (cid:885)](cid:2871)=[(cid:887) (cid:882)(cid:882) (cid:885)](cid:2870)[(cid:887) (cid:882)(cid:882) (cid:885)]=[(cid:887)(cid:2870) (cid:882) If and (cid:1830) exist, they are generally not unique: (cid:2870)=(cid:4666)(cid:1830) (cid:2869)(cid:4667)(cid:4666)(cid:1830) (cid:2869)(cid:4667) (cid:886) (cid:883)], and note that =(cid:1830) (cid:2869) for =[(cid:883) (cid:884: ex. We know =(cid:1830) (cid:2869) and (cid:1830)=[(cid:887) (cid:882) (cid:883) (cid:883)]=[(cid:884) (cid:884) (cid:887)+(cid:884) (cid:885) (cid:887)+(cid:884) (cid:885)] and (cid:2869) do not cancel out; they have an effect. Eigenstuff of a matrix determines whether or not it is diagonalizable. (cid:885) (cid:887) (cid:885) (cid:885) (cid:885: find how many e-vectors has, ie. compute dim(cid:1831) and hope to get 3 linearly independent e, find e-values of :
