= Cholesky decomposition as a sum of squares
{c}
{title2=$u^TAu=\|L^Tu\|_2^2$}
For any real <vector> $u$, a <Cholesky decomposition> gives the <quadratic form> identity
$$
u^TAu=u^TLL^Tu=(L^Tu)^T(L^Tu)=\|L^Tu\|_2^2.
$$
Since $L$ is an <invertible matrix>, this <squared Euclidean norm> is positive whenever $u\ne0$. The factorization thus certifies <positive definiteness> and gives coordinates in which the <quadratic form> is a sum of ordinary squares. For the matrix above,
$$
4u_1^2+4u_1u_2+5u_2^2=(2u_1+u_2)^2+(2u_2)^2.
$$
The <level set> $u^TAu=1$ is an <ellipse>; the <linear map> $z=L^Tu$ sends it to the <unit circle>. The points $u=(1/2,0)^T$ and $u=(-1/4,1/2)^T$ go to $(1,0)^T$ and $(0,1)^T$, respectively.
\Image[/cholesky-quadratic-ellipse-to-circle.png]
{title=A <Cholesky decomposition> turns a positive <quadratic form> into a <squared Euclidean norm>}
{description=Original exact <ellipse> and its image under $z=L^Tu$. Matching point colors show two mapped points; the axes use their own coordinates. This transformation need not be a <rotation> or a uniform scaling.}
The same factor works in the reverse direction for <Gaussian random vectors>. If $Z$ has <independent> unit-<variance>, zero-<mean> <Gaussian random variables> as coordinates, $X=LZ$ has <covariance matrix> $LL^T=A$. In this example $X_1=2Z_1$ and $X_2=Z_1+2Z_2$: their <variances> are $4$ and $5$, while their <covariance> is $2$. This is a concrete way to generate specified correlations from independent draws.
A triangular factor is not generally a <square root of a matrix> in the sense $B^2=A$. Here $L^2=\begin{pmatrix}4&0\\4&4\end{pmatrix}\ne A$. The <Cholesky decomposition> uses the product with the <transpose>, which is also what makes the <quadratic form> a <squared Euclidean norm>.
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