What happens if you multiply a matrix by its transpose?

When you multiply a matrix by its transpose, the result is a symmetric matrix. This means that the resulting matrix will be equal to its own transpose.

To understand this better, let’s consider a matrix A. If we calculate A × A^T (where A^T is the transpose of matrix A), we are taking the dot product of the rows of A with the columns of A^T. Each element in the resulting matrix B will be computed as:

B_ij = Σ (A_ik × A_jk)

This operation results in a matrix where each entry reflects the inner product of the corresponding rows of A. Because of the properties of the dot product, this results in a matrix where B_ij = B_ji, which is the definition of a symmetric matrix.

Additionally, when A is an m x n matrix, the resulting matrix A × A^T is an m x m matrix, having as many rows as A has rows. This operation is widely used in various fields such as statistics and machine learning, particularly in the context of covariance matrices.