Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (**not necessarily idempotent**) 


* what is the distribution of ${y^T M y}$?
* is there a high probability bound on $|{y^T M y}|$?


Most bounds that I could find, such as in <http://www2.econ.iastate.edu/classes/econ500/hallam/documents/QUAD_NORM.pdf> hold only when $M$ is idempotent.



Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix
By the spectral decomposition of $M$, the distribution of $y^T My$ is the same as that of $\sum\_i\sigma^2\mu\_i Z\_i^2$, where the $\mu\_i$'s are the eigenvalues of $M$ and the $Z\_i$'s are iid standard normal random variables (r.v.'s). By rescaling, without loss of generality $\sigma=1$. 


In the paper by [Székely & Bakirov](https://link.springer.com/article/10.1007/s00440-003-0262-6), exact lower and upper bounds on the cumulative distribution function of quadratic forms $Q:=y^T My$ are given. In particular, it is shown that 
$$\inf\_{n\ge1,M\ge0,EQ=1}P(Q\le x)=
\inf\_{n\ge1}P(\chi^2\_n/n\le x)=P(\chi^2\_{n(x)}/n(x)\le x), 
$$
where $x>0$, $n(x)$ depends explicitly and only on $x$, and $\chi^2\_n$ is a r.v. with the chi-squared distribution with $n$ degrees of freedom. 


Among other results, it is also shown in that paper that 
$$\sup\_{M\ge0,EQ=1}P(Q\le x)=P(\chi^2\_n/n\le x)I\{x\ge y(n)\}
+s(x)I\{1\le x<y(n)\}+P(\chi^2\_1\le x)I\{x<1\}, 
$$
where $I$ is the indicator, $x>0$, $y(n)$ depends explicitly and only on $n$, and 
$$s(x):=\sup\_{0\le\mu\le1/2}P(\mu Z\_1^2+(1-\mu)Z\_2^2\le x). 
$$


