Quadratic Forms and the Principal Axis Theorem

Math 303: Section 32

Dr. Janssen


Preview Activity 32.1

Definition 1 Let \(A\) be an \(n\times n\) symmetric matrix. A quadratic form on \(\R^n\) is a function \(Q\) defined by

\[ Q(\x) = \tr{\x} A \x. \]

For a given quadratic form, the matrix \(A\) is unique, so will be referred to as the matrix of the quadratic form.

A question

Given an equation of the form \(Q(\x) = d\), where \(Q\) is a quadratic form on \(\R^2\) and \(d\) is a constant, how can we eliminate the cross terms?

That is, given a quadratic form \(Q\) in variables \(x_1, x_2, \ldots, x_n\), we want to find variables \(y_1, y_2, \ldots, y_n\) in terms of the \(x\)’s so that when written in terms of \(y_1, y_2, \ldots, y_n\), \(Q\) contains no cross terms, i.e.:

\[ Q(\x) = \tr{y} D \y, \]

where \(D\) is diagonal.

Summary: Theorem 32.2

Theorem 1 (Principal Axis Theorem) Let \(A\) be an \(n\times n\) symmetric matrix. There is an orthogonal change of variables \(\x = P \y\) so that the quadratic form \(Q\) defined by \(Q(\x) = \tr{\x} A \x\) is transformed into the quadratic form \(\tr{\y} D \y\), where \(D\) is a diagonal matrix.

Activity 32.1

Let \(Q\) be the quadratic form defined by \(Q(\x) = 2x^2 + 4xy + 5y^2 = \tr{\x} A\x\), where \(\x = \left[\begin{matrix} x \\ y \end{matrix}\right]\) and \(A = \left[\begin{matrix} 2 & 2 \\ 2 & 5 \end{matrix}\right]\).

  • The eigenvalues of \(A\) are \(\lambda_1 = 6\) and \(\lambda_2 = 1\) with corresponding eigenvectors \(\v_1 = \tr{[1 \ 2]}\) and \(\v_2 = \tr{[-2 \ 1]}\), respectively. Find an orthogonal matrix \(P\) with determinant 1 that diagonalizes \(A\). Is \(P\) unique? Is there a matrix without determinant 1 that orthogonally diagonalizes \(A\)? Explain.
  • Use the matrix \(P\) to write the quadratic form without the cross-product.

Classifying Quadratic Forms


A symmetric matrix \(A\) (and its associated quadratic form \(Q\)) is

  • positive definite if \(\tr{\x} A \x > 0\) for all \(\x\ne \mathbf{0}\),
  • positive semidefinite if \(\tr{\x} A \x \ge 0\) for all \(\x\),
  • negative definite if \(\tr{\x} A \x < 0\) for all \(\x\ne \mathbf{0}\),
  • negative semidefinite if \(\tr{\x} A \x \le 0\) for all \(\x\), and
  • indefinite if \(\tr{\x} A \x\) takes on both positive and negative values.

Connecting to eigenvalues

A quadratic form \(Q(\x) = \tr{\x} A \x\) is

  • positive definite if \(A\) has all positive eigenvalues,
  • negative definite if \(A\) has all negative eigenvalues,
  • positive semidefinite if \(A\) has all nonnegative eigenvalues,
  • negative semidefinite if \(A\) has all nonpositive eigenvalues, and
  • indefinite if it has some of both.

Making an inner product

Let \(A\) be a symmetric \(n\times n\) matrix, and define \(\ip{,} : \R^n\times \R^n \to \R\) by

\[ \ip{\u,\v} = \tr{\u} A\v. \]

  • Explain why it is necessary for \(A\) to be positive definite in order for this to define an inner product on \(\R^n\).
  • Show that in this case, the function does define an inner product.
  • Let \(\ip{,} : \R^2 \times \R^2 \to \R\) be defined by

\[ \ip{\left[\begin{matrix} x_1 \\ x_2 \end{matrix}\right], \left[\begin{matrix} y_1 \\ y_2 \end{matrix}\right]} = 2 x_1 y_1 - x_1 y_2 - x_2 y_1 + x_2 y_2. \]

Find a matrix \(A\) so that \(\ip{\x,\y} = \tr{\x} A \y\) and explain why \(\ip{,}\) defines an inner product.