Abstarct: In recent years, extensive applications of matrix functions such as the logarithm and the square root have emerged in interdisciplinary fields like medical image processing. Among these applications, the Log-Euclidean frxamework for medical image registration and analysis is notable, which relies on transforming symmetric positive-definite matrices into a vector space via the matrix logarithm. The accuracy and stability of these practical methods necessitate a precise understanding of the conditions for existence and uniqueness of the logarithm and square root for real matrices. The main objective of this thesis is a systematic investigation of these conditions, leveraging the powerful tool of the real Jordan form. In this regard, the classical Culver’s theorem (1966) on the necessary and sufficient condition for the existence of a real logarithm—recently reconstructed by Gallier (2019) using the real Jordan form—is studied. The key condition is the evenness of the number of Jordan blocks corresponding to each negative real eigenvalue in the matrix structure. Subsequently, by introducing appropriate sets and , the issue of uniqueness for the principal logarithm and the principal square root is resolved. It is demonstrated that the exponential and squaring functions are diffeomorphisms on these sets, respectively. These theoretical foundations underpin stable numerical algorithms such as the Inverse Scaling and Squaring algorithm. Finally, the direct connection of these results to practical applications in medical image processing, especially in non-rigid image registration using the Log-Euclidean Polytopic Transformation (LEPT), is emphasized. This research illustrates how a deep understanding of the algebraic structure of matrices enables the development of reliable and efficient methods in critical domains like medicine.