\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt


QUESTION {\bf Test 1} The characteristic equation of any matrix
$A$ can be written
\begin{equation}
\lambda^n - c_1\lambda^{n-1} + c_2\lambda^{n-2} - \ldots +
(-1)^nc_n = 0.\label{eq:eigen}
\end{equation}

\medskip

If all $c_r$ are positive then all solutions $\lambda$ are
positive.

\medskip

(This is easy to prove: if $\lambda = \alpha < 0$ is a root then
substituting $\alpha = -\beta$ (where $\beta > 0$) into
(\ref{eq:eigen}) gives $(-1)^n \ \times$ a sum of strictly
positive terms, so this cannot be zero.)

\medskip

{\bf Definition} If $A$ is an $n \times n$ matrix then the
following submatrices are called its {\em principal submatrices}:
$$ \left[\begin{array}{l}
 a_{11} \\
\end{array}\right],
\makebox[5mm]{} \left[\begin{array}{ll}
 a_{11} & a_{12} \\
 a_{21} & a_{22} \\
\end{array}\right],
\makebox[5mm]{} \left[\begin{array}{lll}
 a_{11} & a_{12} & a_{13} \\
 a_{21} & a_{22} & a_{23} \\
 a_{31} & a_{32} & a_{33} \\
\end{array}\right],
\makebox[5mm]{} \left[\begin{array}{rrrr}
 a_{11} & a_{12} & a_{13} & a_{14} \\
 a_{21} & a_{22} & a_{23} & a_{24} \\
 a_{31} & a_{32} & a_{33} & a_{34} \\
 a_{41} & a_{42} & a_{43} & a_{44} \\
\end{array}\right],
\makebox{ {\it etc}.} $$

\medskip

{\bf Test 2} A symmetric matrix $A$ is positive definite if and
only if each of its principal submatrices (including $A$ itself)
has positive determinant.

\medskip

{Use each of the two tests above to show that one of the following
matrices is positive definite but the other is not. To what type
of form does the other matrix correspond?}

$$ A = \left[\begin{array}{rrr}
  3 & -1 &  0 \\
 -1 &  2 & -1 \\
  0 & -1 &  3
\end{array}\right],
\makebox[1cm]{} B = \left[\begin{array}{rrr}
  1 & 2 & 1 \\
  2 & 1 & 1 \\
  1 & 1 & 3
\end{array}\right].
$$

\medskip

{For each of the following quadratic forms, use eigenvalues and
eigenvectors to rotate the axes in order to identify what type of
conic it represents.}

\medskip

\hspace{1cm}(a) $9x^2 - 4xy + 6y^2 - 10x - 20y - 5 = 0$,

\hspace{1cm}(b) $3x^2 - 8xy - 12y^2 - 30x - 64y = 0$,

\hspace{1cm}(c) $4x^2 -20xy + 25y^2 - 15x - 6y = 0$,

\hspace{1cm}(d) $9x^2 +12xy + 4y^2 -52 = 0$.



ANSWER $-\lambda^3+8\lambda^2-19\lambda+12=0$ has coefficients
which alternate in sign; the principal subdeterminants are 3,5 and
12. So the quadratic form is positive definite.
$-\lambda^3+5\lambda62-\lambda-7=0$ does not have coefficients
which alternate in sign; the principal subdeterminants are 1,-3
and -7. So the quadratic form is not positive definite, it is
indefinite.


\end{document}