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:

d. millimeters per hour.

step-by-step explanation:

we are given that,

carlo used a rain gauge to measure the amount of water his lawn is receiving from the sprinklers.

since, according to the options, we see that,

water cannot be measured by 'miles per hour', 'meters per second' and 'grams per seconds'.

we get that, the water can be measured by 'millimeters per hour'.

hence, the appropriate rate for describing the amount of water is 'millimeters per hour'.