Matrix times transpose matrix equals identity matrix What are the characteristics of this matrix

Matrix times transpose matrix equals identity matrix What are the characteristics of this matrix

This is the definition of orthogonal matrix
Each column element of the matrix is made into a vector, which is a unit vector, and the column vector groups are orthogonal, so the column vector group is an orthogonal unit vector group. Similarly, the row vector group is also an orthogonal unit vector group
The determinant of a matrix can only be 1 or - 1
Its inverse matrix is its transpose matrix

It is proved that if the transpose of a times a equals zero, then a must be a zero matrix
To be more specific, thank you

Let a = = (a ij) m * n be divided into blocks a = = (A1, A2,..., an), AJ = = (a 1J, a 2J,..., a MJ) (J = = 1,2,... N), then t (a) = = t (t (A1), t (A2),..., t (an))... At (a) = = ∑ AJT (AJ) (J = = 1,2,... N) obviously, AJ is m * 1 matrix, t (AJ) is 1 * m matrix, so at (a) must be m * m matrix