A positive-semi-definite matrix is a type of square matrix in which all of its eigenvalues are non-negative. This means that when the matrix is multiplied by any non-zero vector, the resulting vector will have a non-negative length.
The main difference between positive-semi-definite and positive-definite matrices is that while all eigenvalues of a positive-semi-definite matrix are non-negative, a positive-definite matrix has all of its eigenvalues strictly greater than zero.
Positive-semi-definite matrices are commonly used in fields such as statistics, physics, and engineering. They can be used to represent covariance matrices, which are useful in analyzing data and making predictions. They are also used in optimization problems, such as finding the minimum of a function.
To determine if a matrix is positive-semi-definite, you can check if all of its eigenvalues are non-negative. This can be done by finding the eigenvalues of the matrix using methods such as the Cholesky decomposition or the power method.
Positive-semi-definite matrices have several important properties, including: