An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors. This means that each vector has a length of one and is perpendicular to the others. When an orthogonal matrix is multiplied by its transpose, the result is the identity matrix, denoted as I. This property makes orthogonal matrices useful in various mathematical applications, including solving systems of linear equations and performing rotations in Euclidean space. Orthogonal matrices preserve the length of vectors during transformations, which is essential in fields like computer graphics and signal processing. They are also used in machine learning for dimensionality reduction techniques, such as Principal Component Analysis (PCA), where maintaining the structure of data is crucial.