Semi-definite Matrices

Symmetric square matrices having positive eigenvalues are known as semi-definite matrices. Eigenvalues are found mainly by using the equation; $A - \lambda I=0$ where $A$ is the matrix whose eigenvalues are to be found.

The semi-definite matrix

The definition for a semi-definite matrix; $A$ is:

Some of the observations we can make from this definition are:

1. $-A$ should be negative semi-definite

2. A positive semi-definite matrice, $A$ satisfies the $0 \preceq A$ condition whereas a positive definite satisfies, $0 \prec A$.

Proving positive semi-definite matrices

There are a couple of propositions to look at when defining positive semi-definite matrices.

Non-negative eigenvalues

One of the conditions for a symmetrical matrix to be positive semi-definite is for it to have all eigenvalues as positive:

In this proposition, $A$ is the matrix while $s$ is a vector. This has been decomposed to create further equations within the definition. We can notice that the expression is positive for all $s$ if all of the $\lambda$ values from $i,...,n$ are positive.

Matrix decomposition with similar rank

Another proposition states that having a matrix, such as$A$ can be broken into one other matrix, $V$ that would consist of all the eigenvectors of A, basically corresponding to the positive eigenvalues:

The point to be noted here is that if such a decomposition is possible, only then will a symmetrical matrix be positive semi-definite.

Semi-definite matrices can be both positive and negative. Taking into account the importance of eigenvalues and eigenvectors, matrices can be defined further into different subcategories.