Hessian Matrix Test - When does it fail?

Question

When does the hessian matrix test fail. I understand we are testing the definiteness of the Matrix, and i also understand that because it's a symmetric $n•n$ matrix, we have a principal minor conditions for semi-definitness that we don't have with a non-symetric matrix. Please point out any errors i have made, answer the individual questions if you have time, and add anything you think i have missed. Thanks!

Negative Definite

1. Principal minors Alternate in Sign with the first one being Negative.

• Note: For a $2•2$ matrix we can use either $h_{11}$ or $h_{22}$ as our first principal minor.
• Question: What if $h_{11}$ and $h_{22}$ have alternate signs?
• Question: How does this scale up to larger matrices. If so is it h_{nn} we can use as our first principal minor?

2. Eigen values are all negative. This suggests that the trace will be negative as well?
3. These conditions mean our function is Strictly Concave.

Negative Semi-Definite

1. Principal minor conditions above but the Determinant can also be 0.

• Question: If our symmetric Matrix is grater than $2•2$ can more than one principal minor be equal to 0?

2. Eigen Value conditions above but one (or more?) Eigenvalues can = 0

• Question: Can more than one eigenvalue = 0 for $n•n$ where $n>2$?

3. Function is concave.

Positive Definite

1. Principal minors > 0.

• Again for a $2•2$ matrix we can use either $h_{11}$ or $h_{22}$ as our first principal minor.
• Question: For a $2•2$ matrix what if $h_{11}$ is positive and $h_{22}$ negative?
• Question: Again, how does this scale up to larger matrices.

2. Eigen values are all Positive?
3. These conditions mean our function is Strictly Convex.

Positive Semi-Definite

1. Principal minor conditions above but the Determinant can also be 0.

• Question: Same as before - If our symmetric Matrix is grater than $2•2$ can more than one principal minor be equal to 0?

2. Eigen Value conditions above but one (or more?) Eigenvalues can = 0

• Question: Can more than one eigenvalue = 0 for $n•n$ when $n>2$?

3. These conditions mean our function is convex.

Summary Questions

1. When does the hessian test for a symmetric Matrix fail?
2. More generally when does the tests on our Jacobin fail. E.g. say we are linearising around a fixed point in a system of differential equations ($\dot{x}, \dot{y}$), and so our second derivative in the Taylor series is the Jacobian. When we test the definitness of the Jacobian, under what conditions does this fail?
3. Looking at the example $f(x,y) = 3xy$ the hessian for this example is.

• $H =\begin{bmatrix} 0 & 3 \\\ 3 & 0 \end{bmatrix}$
• Here we have $h_{11} = h_{22} = 0$ i have been told previously, that this function is quasiconcave but not concave. So this an example of the Hessian test failing?

4. Are the statements above for Concavity and Convexity iff statements. I.e. If the function is Strictly Concave, it's hessian will be Negative Definite?