Question

Please help me step by step and as clearly as possible PLEASE, Please treat me like an absolute idiot and list out every step by step please. I got this home work that says the following:

Find all Jordan Canonical forms of a 3x3 matrix A with. ( what does it mean when saying I can assume that the) diagonal entries are in increasing order.

det(A – xIq) = (2 – x)(1 – x)2.

Answer 1

Jordan Canonical form & If de matrix Anonhas characteristic equation, polynomial CA (A) = (x-A)" (andjo then, Jordan will be of the form, (ardn) O O - O do non ti here, If minimal polynomial MACA ma (A) = (x-2)(x-t ..... (x-dinjam here mi chi then, Jordan form of matux Anxn contains: h atleast one jordan block of order me with respect to eigen value de. Jordan block: A block matin which contains a eigen value & as a digenoal and it has element 1 as sperdiagonal (above the main diagonal. eg. оо odio Jordan Block for each eigen value i.

In our question : det (A-x Ia) = (2-x3 (1-xp. 5) Characteristic, equation (co () = (dame)? (2x-1) ? polynomial since minimal polynomial divides characteristic polynomial and contains att factor of each eigen value therefore, minimal polynomial can be [masal= (x-2)(x-1 [MA (A) = (1-25 (x-1) (mA (A)] = (x-2) a (x-1) ? [MA (A)} = (x-2) (x-15 eigen value 1 there are only four possibilities. so let us make jordan form of each type of minimal polynomial with diagonal entries in increasing order, ie first we write block of and than jordans block of eigen value 2, when [mA (d)], where (mA (A) [MA (A) (mod) [[ uo 3: olilo 0 (0) 0 [2] ooo [2] [[100 O 0 0 o 0 оо 0 02 0 O 0 0 02 0 00012 O 0 0 0 0 Ano.

Esplanation about [ma (dl), (mA, - (2-2)! (2-1) by first sule (sule Opohrich be we mentioned earty), jordan forms will contains atleast a Jordans block of eigen value & of order 2 & contain atleast one jordan block of order i of eigen value 1. Now, by characteristic polynomial, we that, eigen values are so if we make a 242 jordan block of eigen value 2, then we have not more eigen values to make new fordan blocks of order less than can say 1 two. hence, jordan from for EMAN = 0 Co 0 21 make all four jordan forms Similarly, we can of matrix A fos given characteristic. Remember: o find possible all minimal polynomial. by applying rike (mentioned at first page), find all possible jordan forms.