Absence of cross product terms in quadratic forms.

Question

Question is whether the following statement is true or false?

if the quadratic form $X^T AX$ has no cross product terms then $A$ is diagonal matrix.

I know that, if A is diagonal matrix then quadratic form has no cross product terms! But what if the quadratic form has no cross product terms? Is A will be diagonal matrix?

If there are no cross product terms in quadratic form then all off diagonal entries of matrix are zeros. So A will be diagonal. But in key it is given that, answer is false! That is $A$ is not diagonal! Please explain? and if possible please give me an examples.

Answer 1

Hint: $$\begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} a & b \\ -b & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} =?$$

Answer 2

if the quadratic form ${x}^T{Ax}$ has no cross product terms then A is a diagonal matrix.

This statement is false because, according to The Principal Axes Theorem, $x^TAx$ can be expressed as $y^TDy$. This requires that we change variable in this manner: $$ x=Py\:\text{ so }\:{x^T}=y^TP^T $$ thus $x^TAx = y^TP^TAPy$.

Now, in this particular scenario $P$ is an orthogonal matrix ($P^{-1} = P^T$) that orthogonally diagonalizes A i.e. $P^TAP = D$, and we have that $$ y^TP^TAPy = y^TDy $$