Diagonal Matrix

Chapter 2: Matrix. In the chapter mind map this icon labels Transpose ($A^\top$) & Diagonal Matrices. The discussion below is excerpted and lightly edited from § Transpose in Mathematics for AI and Machine Learning. Related material also appears in Chapter 5 (Trace & Cyclic Props), Chapter 5 (Determinants & Inverses via LU).

The transpose operation flips a matrix along its main diagonal, converting rows into columns and columns into rows. For a matrix $A \in \mathbb{R}^{M \times N}$, the transpose $A^\top$ is an $N \times M$ matrix where the element at position $(i,j)$ in $A$ becomes the element at position $(j,i)$ in $A^\top$. This operation is fundamental in linear algebra, appearing frequently in matrix operations, inner products, and the formulation of symmetric matrices.

and transposing twice restores the original matrix, i.e., $(A^\top)^\top = A$.