Is the Matrix Positive-Definite? Introduction
Today we're gonna take care of the bane of first-year linear algebra students: matrix definiteness! Apparently this doesn't yet have a challenge so here we go:
Input

A \$n\times n\$ symmetric Matrix \$A\$ in any convenient format (you may also of course only take the upper or the lower part of the matrix)
Optionally: the size of the matrix \$n\$

What to do?
The challenge is simple: Given a real-valued matrix \$n\times n\$ Matrix decide whether it is positive definite by outputting a truthy value if so and a falsey value if not.
You may assume your built-ins to actually work precisely and thus don't have to account for numerical issues which could lead to the wrong behaviour if the strategy / code &quot;provably&quot; should yield the correct result.
Who wins?
This is code-golf, so the shortest code in bytes (per-language) wins!

What is a positive-definite Matrix anyways?
There are apparently 6 equivalent formulations of when a symmetric matrix is positive-definite. I shall reproduce the three easier ones and reference you to Wikipedia for the more complex ones.

If \$\forall v\in\mathbb R^n\setminus \{0\}: v^T Av&gt;0\$ then \$A\$ is positive-definite. This can be re-formulated as: If for every non-zero vector \$v\$ the (standard) dot product of \$v\$ and \$Av\$ is positive then \$A\$ is positive-definite.
Let \$\lambda_i\quad i\in\{1,\ldots,n\}\$ be the eigenvalues of \$A\$, if now \$\forall i\in\{1,\ldots,n\}:\lambda_i&gt;0\$ (that is all eigenvalues are positive) then \$A\$ is positive-definite. If you don't know what eigenvalues are I suggest you use your favourite search engine to find out, because the explanation (and the needed computation strategies) is too long to be contained in this post.
If the Cholesky-Decomposition of \$A\$ exists, i.e. there exists a lower-triangular matrix \$L\$ such that \$LL^T=A\$ then \$A\$ is positive-definite. Note that this is equivalent to early-returning &quot;false&quot; if at any point the computation of the root during the algorithm fails due to a negative argument.

Examples
For truthy output
\begin{pmatrix}1&amp;0&amp;0\\0&amp;1&amp;0\\0&amp;0&amp;1\end{pmatrix}
\begin{pmatrix}1&amp;0&amp;0&amp;0\\0&amp;2&amp;0&amp;0\\0&amp;0&amp;3&amp;0\\0&amp;0&amp;0&amp;4\end{pmatrix}
\begin{pmatrix}5&amp;2&amp;-1\\2&amp;1&amp;-1\\-1&amp;-1&amp;3\end{pmatrix}
\begin{pmatrix}1&amp;-2&amp;2\\-2&amp;5&amp;0\\2&amp;0&amp;30\end{pmatrix}
\begin{pmatrix}7.15&amp;2.45\\2.45&amp;9.37\end{pmatrix}
For falsey output
(at least one eigenvalue is 0 / positive semi-definite)
\begin{pmatrix}3&amp;-2&amp;2\\-2&amp;4&amp;0\\2&amp;0&amp;2\end{pmatrix}
(eigenvalues have different signs / indefinite)
\begin{pmatrix}1&amp;0&amp;0\\0&amp;-1&amp;0\\0&amp;0&amp;1\end{pmatrix}
(all eigenvalues smaller than 0 / negative definite)
\begin{pmatrix}-1&amp;0&amp;0\\0&amp;-1&amp;0\\0&amp;0&amp;-1\end{pmatrix}
(all eigenvalues smaller than 0 / negative definite)
\begin{pmatrix}-2&amp;3&amp;0\\3&amp;-5&amp;0\\0&amp;0&amp;-1\end{pmatrix}
(all eigenvalues smaller than 0 / negative definite)
\begin{pmatrix}-7.15&amp;-2.45\\-2.45&amp;-9.37\end{pmatrix}
(three positive, one negative eigenvalue / indefinite)
\begin{pmatrix}7.15&amp;2.45&amp;1.23&amp;3.5\\2.45&amp;9.37&amp;2.71&amp;3.14\\1.23&amp;2.71&amp;0&amp;6.2\\3.5&amp;3.14&amp;6.2&amp;0.56\end{pmatrix}
C, 108 bytes

-1 byte thanks to Logern
-3 bytes thanks to ceilingcat



f(M,n,i)double**M;{for(i=n*n;i--;)M[i/n][i%n]-=M[n][i%n]*M[i/n][n]/M[n][n];return M[n][n]&gt;0&amp;(!n||f(M,n-1));}


Try it online!

Performs Gaussian elimination and checks whether all diagonal elements are positive (Sylvester's criterion). Argument n is the size of the matrix minus one.