03_003_Inverse_matrix_Positive-definite_Positive-semidefinite_matrix.html ========================================================= inverse number $$$3\times \frac{1}{3}=1$$$ $$$\frac{1}{3}$$$ is inverse number of 3 inverse matrix $$$AA^{-1}=I$$$ $$$A^{-1}$$$ is inverse matrix of A ========================================================= Characteristic of inverse matrix $$$AA^{-1}=I$$$ $$$A^{-1}A=I$$$ $$$(A^{-1})^{T}=(A^{T})^{-1}$$$ $$$(AB)^{-1}=B^{-1}A^{-1}$$$ ========================================================= positive-definite is characteristic of matrix which has similar characteristic with positive real number (number)$$$\times$$$ (positive number) $$$\rightarrow$$$ changes size but +- doesn't change In matrix, there are matrices which have above characteristic, because matrix is also multiplied by some others Matrix can play role of positive number (positive-definite) or negative number (negative-definite) ========================================================= Mathematically, if A satisfies $$$x^{T}Ax>0$$$, A is positive-definite matrix Simply, you can write A>0 to express A is positive-definite matrix ========================================================= example of positive-definite matrix : $$$A=\begin{bmatrix} 3&0&0 \\ 0&1&0 \\ 0&0&2 \end{bmatrix}$$$ ========================================================= If A satisfies $$$x^{T}Ax \geq 0$$$, A is positive-semidefinite matrix Famous example of positive-semidefinite matrix is covariance matrix