Suppose that X and A are matrices and A is positive definite. (A is a positive definite matrix if **x’**A**x** > 0 for every nonzero vector **x**.) The matrix X is called the square-root of A if X^{2} = A. Here, X^{2} is the multiplication of X by itself, that is X^{2} = X⋅X. Furthermore, the square-root of A is denoted by A^{1/2}. How to find the square-root of a matrix? This article addresses the problem.

Let A be a k × k positive definite matrix with the following spectral decomposition.

where are eigenvalue-eigenvector pairs of A with the properties that all eigenvectors have norm 1 and they are orthogonal to each other. By defining the matrix and the matrix where , the decomposition can also be expressed as:

It can be proved that the matrix defined by is a solution to the matrix equation X^{2} = A. (In other words, A^{1/2}⋅A^{1/2} = A.) Additionally, if we define the matrix where , then A^{1/2} can also be expressed as:

Some of the properties of the square-root matrix are as follows.

- A
^{1/2}is a symmetric matrix. - The inverse of the matrix A
^{1/2}, which is and denoted by A^{-1/2}, can be determined by the following formula: where and . - The square of A
^{-1/2}is the inverse of A. In other words,

**Example**

Given a positive definite matrix , find A^{1/2}, A^{-1/2}, and A^{-1}.

**Answer**

The eigenvalues of A are λ_{1} = 6 and λ_{2} = 1. For λ_{1} = 6, we get and for λ_{2} = 1 we have . From , λ_{1}, and λ_{2} we get the matrices and . As a consequence, the spectral decomposition of A is as follows.

To find A^{1/2}, the first step is to calculate Λ^{1/2}, that is . This results in the following square-root matrix.

To determine A^{-1/2}, the first step is to compute Λ^{-1/2}, i.e. . Therefore, we obtain the square-root matrix:

To determine A^{-1}, we can use the formula above: A^{-1/2} ⋅ A^{-1/2} = A^{-1}

It follows that

**Note about finding A ^{-1}**

Indeed there are other ways to determine A^{-1}, but in this article we have deliberately applied just to demonstrate how to use the formula. Furthermore, if A has the spectral decomposition A = PΛP’, its inverse can also be expressed as or A^{-1} = PΛ^{-1}P’ where and .

If A^{-1} in the example above is determined using the formula A^{-1} = PΛ^{-1}P’, then and it follows that: