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shaon0
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Homework Statement
See Attachment.
Homework Equations
x^TAx where x=<x,y,z> and A is some 3x3 matrix
The Attempt at a Solution
See Attachment. From here i have found the eigenvalues and vectors. What should I do next?.\
shaon0 said:Homework Statement
See Attachment.
Homework Equations
x^TAx where x=<x,y,z> and A is some 3x3 matrix
The Attempt at a Solution
See Attachment. From here i have found the eigenvalues and vectors. What should I do next?.\
Ray Vickson said:I would have not bothered at all to get the eigenvalues; I would just perform Cholesky factorization on A: find an upper-triangular matrix U such that A = U^T * U. Just expand this out to find the following algorithm for U:
U[1,1] = sqrt(A[1,1]) and for j=2,...,n, U[1,j] = A[1,j]/U[1,1].
for i from 2 to n do U[i,i]=sqrt(A[i,i]-sum_{k=1..i-1} U[k,i]^2) and for j=i+1,...,n,
U[i,j] = (A[i,j] - sum_{k=1..i-1} U[k,i]*U[k,j])/U[i,i].
(Note: even if you have not seen this before, it only takes about 5-10 minutes to master, and then you can do it quickly by hand for matrices up to about 10x10 or even a bit more, using nothing beyond a hand-held calculator.)
For your matrix A we have U = [[1,1,2],[0,sqrt(3),4sqrt(3)],[0,0,2]] (= [row1,row2,row3])
So, if u1 = (row1 of U) * <x,y,z> = x+y+2z, u2 = (row2 of U) * <x,y,z> = sqrt(3)y + 4sqrt(3)z and u3 = (row3 of U) * <x,y,z> = 2z, then Q = x^T A x = u1^2 + u2^2 + u3^2. If you expand this out you will see that you get back to your original form, so it gives you exactly what you want: Q written as a sum of squares. No eigenvalues need be involved at all.
RGV
The purpose of solving homework equations is to find the value or values of the variable(s) that make the equation true. This allows for a better understanding of the concept being taught and helps to reinforce problem-solving skills.
x^TAx is a mathematical notation used to represent a quadratic form, where x is a vector and A is a matrix. The equation is solved by finding the values of x that satisfy the equation.
x^TAx is different from other types of equations because it involves vectors and matrices, which may have multiple variables and require different methods of solving. It also often involves finding the minimum or maximum value of the equation rather than a specific solution.
Some common strategies for solving homework equations include factoring, substitution, completing the square, and using the quadratic formula. It is also important to carefully analyze the given equation and identify any patterns or properties that may be helpful in solving it.
You can check if your solution to a homework equation is correct by plugging the values you found for the variable(s) into the original equation and seeing if it satisfies the equation. You can also use a graphing calculator or online graphing tool to visually confirm the solution. Additionally, you can compare your solution to the solution provided by the teacher or textbook.