# Decomposition of a symmetric matrix into multiplication of two vectors

What is the necessary condition for a real symmetric matrix $A_{m\times m}$ to be written as $B*B^T$ where $B$ is an $(m\times 1)$ matrix ?

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$B*B^T$ will have rank 1 – Simon Markett Jun 1 '12 at 9:00
then it is not possible unless $A$ has only one independent row/column. Thanks! – Tarek Jun 1 '12 at 9:12
Consider answering your own question and accepting it. – Inquest Jun 1 '12 at 10:20
Also relevant to see: The matrix will be semi-positive definite. Also, the numbers on the diagonal have to have squareroots. – sebigu Jun 1 '12 at 15:19
How about the more general case $A=B*C$ where $C$ is a $1\times m$ matrix and $A$ is a general matrix ? – Tarek Jun 2 '12 at 9:25

How about the more general case $A=B∗C$ where $C$ is a $1×m$ matrix and $A$ is a general matrix ?
Take $A={\vec B}^T\vec C+{\vec C}^T\vec B$. Then $A$ is symmetric, due to $$A^T=\left({\vec B}^T\vec C+{\vec C}^T\vec B\right)^T=\left({\vec B}^T\vec C\right)^T+\left({\vec C}^T\vec B\right)^T={\vec C}^T\vec B+{\vec B}^T\vec C=A$$