Consider the quadratic form q(x, y,z)=11x^2-16xy-y^2+8xz-4yz-4z^2. find an orthogonal change of variable that eliminates the cross product in q(x, y,z) and express q in the new variables.

Step-by-step explanation:

The given quadratic form is of the form

.

Where .Every quadratic form of this kind can be written as

Observe that is a symmetric matrix. So is orthogonally diagonalizable, that is to say,   where is an orthogonal matrix and is a diagonal matrix.

In our case we have:

The eigenvalues of are .

Every symmetric matriz is orthogonally diagonalizable. Applying the process of diagonalization by an orthogonal matrix we have that:

Now, we have to do the change of variables to obtain

Which can be written as:

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step-by-step explanation:

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