Question

Let q(x, y) =X²-6xy-7y find an orthogonal substitution that diagonilizes q​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

You have to find the Matrix equivalent of the quadratic equation. This is a very detailed workout!

Step-by-step explanation:

q(x,y) = x² - 6xy-7y² (You copied the question wrong]

the matrix for this will be A  [ δ²q / δx²   δ²q / δxδy

                                                δ²q / δxδy   δ²q / δy² ]

where δ signifies partial derivation.

hence δq/δx = 2x - 6y --- (i)

δ²q / δx² is differentiating (i) with x again  = 2

δ²q / δxδy is differentiating (i) with y = -6

similarly ;

δ²q / δxδy = -6

and

δ²q / δy² = -7

hence A = [  2   -6

                    -6    -7]

let λ be the Eigen Values | A- λIₙ | = 0

⇒det( [ 2-λ   -6

      -6    -7-λ ] )  = 0

⇒ λ²-5 λ -50 =0

⇒ λ =[ -(-5)±√(25-4*1*50) ] / 2*1 ( Using Sreedhar Acharya Method. { Sreedhar Acharya is an Indian Mathematician who developed a way for quickly solving quadratics; ax²+bx+c ⇒ x =  [-b±√(b² - 4ac)] / 2a; this is always accurate. }

= (5±15) / 2 ⇒ λ = -5, 10

Hence D = [ -5  0

                     0   10]

We now find orthogonal eigen vectors ; say v1,v2;

P = [ v1  v2]

P ⁻¹ is the inverse of P;

then D = P⁻¹AP

and you can show A = PDP⁻¹

( I assume you know how to find eigen values and eigen vectors since you have a doubt with diagonalization; which is a lot advanced).