Question

Determine whether A is diagonalizable. If A is not diagonalizable, explain why nit. If A is diagonalizable, find an invertible matrix P and a diagonal matrix D such that P'AP=D

Answer 1

Diagonalization of a matolx Two nxn mataices A and BB are similar if and only if there is a non singular mataine P sub that no ļ De B=Paipt The algebou of matorces applings applies smoothly to diagonal matrices. To add Cor multiply) any two diagonal matorkes one simply add Coo multiply) cordesponding diagonal entries. For this and other season it is important to know which matrices are similar to diagonal matolces and which pairs olodiagonal matalces are similar to each other Theodem If a square malix of loader n has a linearly independent eiges vectors, then a mataix can be APAR found such that Rod in a diagonal matar.

Proof! 1. For simplicity, we prove the theraers for a squase malaix of order 3. The prool can be easily extended to maldices of higher codess. Let 193, age Qas be a square mataix of cader 3.. Let ?, ?, hg be the eigen values of A and *, , xx, xs the coadesponding eigen vectoos. Let; Since eigen vectors are non. Bero solutions oë the equation Ax=RX, we have . PX;= 2, X., PX = ?X and A % 2 2,73 het .

مي 50 N 2, 3 consider APB A , & x] - [AX, AX Ars] -Tax. 2,3] (2,2 222 2²3) o ů s o 2 N where SPD where Da diag (2,3,7) 03 ApPD. multiplying both sides by B BAP - ppo ID=D Doe

1. The matar dep which diagonalises à is called the modal mateix and the desulting diagonal mateix Dis known as the spectaal matalx A. d. The transformation of a malaix A to pap. is known as a similarity to ansformation A. Annan matalx î is so called similao to as nyn' mahoix A is Â. B' Ap for some (non. singular) nan matorx po Examples Diagonalise the state's more ] 12 31 solation The characheolstic equation of A. is (A-21) 20 is for welcom le. I and I

he, (4-2)(3-2)-2220 je, g -78 +10=0 (2-2)(2-5) -0 => 224,5 , when had, the eigen verdoo X is given by CA-21)x=0 => 2x+y=o=)dx=-ya Thus the eigen vector codzespond.cy to Asd is x,-1:7 when 225, the eigen vector is given by (A-31) X=0. ich es ] [51-0-2-9][3]. which gives only one independrech equation asety zo ie, stay in a Thus the eigen rector Coogesponding to R25 is 42:7 Let, B[1, x].[ Ja sockel maliza 2.

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