Write the matrix associated with the quadratic form \(Q(x,y)=3x^2+5y^2+2xy\).
Write the symmetric matrix for \(Q(x,y,z)=x^2+2y^2+3z^2+2xy+4xz+6yz\).
For \(Q(x,y)=4x^2+9y^2-12xy\), write the corresponding matrix and show it is singular.
Express \(Q(x,y)=2x^2+2xy+y^2\) as \(\mathbf{x}^T A\mathbf{x}\) and give matrix \(A\).
Find the matrix for \(Q(x,y,z)=x^2+y^2-z^2\) and state its trace and determinant.
Convert the following matrix into quadratic form:
\[
A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}
\]
Find the quadratic form \(Q(x) = x^T A x\).
Convert the following matrix into quadratic form:
\[
B = \begin{bmatrix} 4 & -2 \\ -2 & 5 \end{bmatrix}
\]
Find the quadratic form \(Q(x) = x^T B x\).
Convert the following matrix into quadratic form:
\[
C = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 0 \\ 1 & 0 & 2 \end{bmatrix}
\]
Find the quadratic form \(Q(x) = x^T C x\).
A. Reduction to Canonical (Diagonal) Form
Reduce \(Q(x,y)=7x^2+2xy+3y^2\) to diagonal form using an orthogonal transformation (show steps).
Show that \(Q(x,y)=4x^2+12xy+9y^2\) reduces to a perfect square and give the canonical form.
Reduce \(Q(x,y,z)=2x^2+2y^2+2z^2+2xy+2xz+2yz\) to diagonal form by finding eigenvectors.
Find an orthogonal matrix that diagonalizes \(Q(x,y)=13x^2-10xy+13y^2\) and write the canonical form.
Reduce \(Q(x,y)=9x^2+12xy+4y^2\) into canonical form and identify the change of variables.
B. Canonical Form — Methods (Completing the Square / Orthogonal)
Use completing the square to reduce \(Q(x,y)=x^2-4xy+4y^2\) to canonical form.
Apply a rotation (orthogonal transformation) to diagonalize \(Q(x,y)=5x^2+6xy+5y^2\).
Reduce \(Q(x,y,z)=x^2+y^2+z^2+2xy\) to diagonal form and give the new variables.
Find canonical form of \(Q(x,y)=6x^2+8xy+3y^2\) by computing eigenvalues of its matrix.
Diagonalize \(Q(x,y,z)=x^2+y^2+z^2-2xy-2yz-2xz\) and list the diagonal entries.
C. Definiteness Classification (Positive / Negative / Indefinite)
Classify \(Q(x,y) = x^2+2xy+y^2\) (use Sylvester's criterion or eigenvalues).
Classify \(Q(x,y) = -x^2-2xy-y^2\).
Classify \(Q(x,y,z)=x^2+y^2+z^2\).
Classify \(Q(x,y)=x^2-2xy+y^2\) and explain whether it is positive semidefinite or definite.
Classify \(Q(x,y)=x^2-y^2\) and state its index and signature.
Classify \(Q(x,y,z)=x^2+2y^2-z^2\).
Decide definiteness of \(Q(x,y)=3x^2+2xy+3y^2\) by Sylvester's criterion.
Decide definiteness of \(Q(x,y)=2x^2-4xy+2y^2+z^2\) and justify.
Is \(Q(x,y,z)=(x+y+z)^2\) positive definite, semidefinite, or indefinite? Explain.
Show whether \(Q(x,y)=-4x^2-4xy-y^2\) is negative definite, semidefinite, or indefinite.
D. Computation & Applied Exercises
Given matrix \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\), write the quadratic form \(\mathbf{x}^T A\mathbf{x}\) and classify it.
For matrix \(B=\begin{pmatrix}1&2&1\\2&4&2\\1&2&1\end{pmatrix}\), find the rank and reduce the associated quadratic form to canonical form.
Show that \(Q(x,y)=2x^2+2xy+y^2\) can be written as a sum of squares and determine definiteness.
Find the principal axes for \(Q(x,y)=7x^2+2xy+3y^2\) (i.e., eigenvectors) and normalize them.
Given \(Q(x,y,z)=2x^2+5y^2+3z^2+4xy-2xz\), construct the symmetric matrix and state whether it is positive definite.
Prove that if the symmetric matrix of a quadratic form has all positive eigenvalues then the form is positive definite.
Find an example of a 2×2 quadratic form that is indefinite and explain why.
Show that the quadratic form corresponding to \(A=\begin{pmatrix}0&1\\1&0\end{pmatrix}\) is indefinite by finding a vector \(\mathbf{x}\) with negative value and one with positive value.
For \(Q(x,y)=9x^2+12xy+4y^2\) find the orthogonal matrix that diagonalizes it and give the diagonal form.
Given \(Q(x,y)=x^2+4xy+4y^2\), reduce it to canonical form and classify definiteness.
