# Positive-definite matrix

• The matrix $M_0 = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$ is positive definite. To prove this, we choose a vector with entries $\textbf{z}= \begin{bmatrix} z_0 \\ z_1\end{bmatrix}$. When we multiply the vector, its transpose, and the matrix, we get: $\begin{bmatrix} z_0 & z_1\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} z_0 \\ z_1\end{bmatrix}=\begin{bmatrix} z_0\cdot 1+z_1\cdot 0 & z_0\cdot 0+z_1\cdot 1\end{bmatrix} \begin{bmatrix} z_0 \\ z_1\end{bmatrix}=z_0^2+z_1^2;$
when the entries z0, z1 are real and at least one of them nonzero, this is positive. This proves that the matrix $M_0$ is positive-definite.