Let $A \in \mathbb{R}^{2 \times 2}$ with
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
I want to know how one can prove/disprove if
$1.$ A is positive-definite if $a > 0$ and $\det(A) > 0$
$2.$ A is negative-definite, if $a < 0$ and $\det(A) > 0$
$3.$ A is indefinite if $\det(A) < 0$
I know that a symmetrical matrix $A \in \mathbb{R}^{n \times n}$ is
- positive definite if $(x,Ax) > 0$ for all $x \in \mathbb{R^{n\times n} \{ 0\}}$
- negative definite if $-A$ is positive definite
- indefinite if vectors $x,y \in \mathbb{R}^{n \times n}$ exist with $(x,Ax) > 0$ and $(y,Ay) < 0$
I think $1.$ is true, because if we look at the canonical forms:
$$
\begin{pmatrix}1&0\\0&1\end{pmatrix},\,
\begin{pmatrix}-1&0\\0&-1\end{pmatrix},\,
\begin{pmatrix}1&0\\0&0\end{pmatrix},\,
\begin{pmatrix}-1&0\\0&0\end{pmatrix},\,
\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
the matrix $A$ shares with its canonical form the sign of the determinant (including being $0$). So, $\det(A)>0$ immediately gives $A$ definite, and it remains to distinguish whether $A$ is positive or negative. In any case, the two entries in the diagonal of $A$ have the same sign, hence the sign of their sum, which is the trace of $A$. Thus $\det(A)>0$, tr$(A)>0$ means positive definite.
Regarding $2.$ I believe it's false, because $a_{11}$ has to be $< 0$ and not $> 0$. Counterexample:
\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
$a$ is $<0$ and $\det(A)>0$ but it's positive definite
I think $3.$ is true because a matrix is indefinite if there are positive and negative eigenvalues. The formula for $\det A$ is
$$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc$$
For example
$$\begin{vmatrix} a & 0 \\ 0 & d \end{vmatrix} = a \cdot d -0 = ad$$ and for $\det < 0$ it is necessary that $a$ or $d$ are negative. (But since this is an example it's not a proof, right? …)
Is this correct?
Best Answer
Let $ B=(A+A^T)/2 $ .Then, for any 2-element row-vector v, $$ vAv^T=vBv^T $$ and Bis symmetric, so A is positive definite, negative definite, indefinite iff the same is true for B. The upper-left corner of B is a, the lower right corner of B is d and the other element is (b+c)/2. So positive definieness of A or B means $$ a>0, ad-(b+c)^2/4>0 $$ negative definiteness of A or B means $$ a<0, ad-(b+c)^2/4>0 $$ and indefiniteness of A or B means $$ ad-(b+c)^2/4<0 $$ns