using one vector to build positive definite matrix

Question

Let $X=(X_1,X_2,...,X_n)_{1\times n}$ be a n-dimensional vector and a matrix $A_{n\times n}=(X^{T})_{n\times 1} * X_{1\times n}$. Under what condition of $X$, $A$ is a (semi-)positive definite matrix?

Many thanks!

Answer 1

The matrix will be positive semidefinite. To see this multply $A$ from the left and right by $Y$ and $Y^T$ for row vectors $Y$. The result will be $(XY^T)^2$.

It will be positive definite if and only if $X\ne0$ and $n=1$. I.e., the matrix is not positive definite if there is a non-zero vector $Y$ orthogonal to $X$.