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Show the following matrix is diagonalizable

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Asked by aryansohanyadav205 26th November 2022, 9:20 PM
Answered by Expert
Answer:
(i) To prove A is diagonalizable where A = [-2  -8  -12 // 1  4  4 // 0  0  1]
Consider, A - lambda I = 0
open vertical bar straight A minus λI close vertical bar equals open vertical bar open square brackets table row cell negative 2 end cell cell negative 8 end cell cell negative 12 end cell row 1 4 4 row 0 0 1 end table close square brackets minus open square brackets table row straight lambda 0 0 row 0 straight lambda 0 row 0 0 straight lambda end table close square brackets close vertical bar equals open vertical bar table row cell negative 2 minus straight lambda end cell cell negative 8 end cell cell negative 12 end cell row 1 cell 4 minus straight lambda end cell 4 row 0 0 cell 1 minus straight lambda end cell end table close vertical bar
equals left parenthesis negative 2 minus straight lambda right parenthesis left parenthesis 4 minus straight lambda right parenthesis left parenthesis 1 minus straight lambda right parenthesis plus 8 left parenthesis 1 minus straight lambda right parenthesis
equals left parenthesis negative 8 minus 4 straight lambda plus 2 straight lambda plus straight lambda squared plus 8 right parenthesis left parenthesis 1 minus straight lambda right parenthesis
equals left parenthesis straight lambda squared minus 2 straight lambda right parenthesis left parenthesis 1 minus straight lambda right parenthesis
equals straight lambda left parenthesis straight lambda minus 2 right parenthesis left parenthesis 1 minus straight lambda right parenthesis

open vertical bar straight A minus λI close vertical bar equals 0 space
rightwards double arrow straight lambda equals 0 comma space 1 comma space 2
Take space different space values space of space straight lambda space in space left square bracket straight A minus λI right square bracket open square brackets table row cell straight x subscript 1 end cell row cell straight x subscript 2 end cell row cell straight x subscript 3 end cell end table close square brackets equals open square brackets table row 0 row 0 row 0 end table close square brackets
Find space values space of space eigen space vectors space straight x subscript 1 comma space straight x subscript 2 comma space straight x subscript 3

Form space straight a space matrix space straight P space with space eigen space vectors space straight x subscript 1 comma space straight x subscript 2 comma space straight x subscript 3 space where space straight x subscript 1 comma space straight x subscript 2 comma space straight x subscript 3 space represents space the space three space different space columns.
If space straight P to the power of negative 1 end exponent AP equals straight D space left parenthesis diagonal space matrix right parenthesis space then space straight A thin space is space diagonalizable.
Answered by Expert 19th December 2022, 9:55 AM
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