The transposition of the 1. matrix is equal to its inverse matrix. That is, if a is an n×n matrix, then the transpose of a (denoted as at) is equal to the inverse matrix of a (denoted as A- 1). Mathematically expressed as t = a- 1.
2. All column vectors of the matrix are unit vectors and orthogonal to each other. This means that for any two different column vectors v_i and v_j, their dot products are all zero, that is, v _ i v _ j = 0. Meanwhile, the module length of each column vector is 1, that is |||| v _ i |||| =1.
To prove that a matrix is an orthogonal square, we can follow the following steps:
1. First, we need to verify whether the transpose of a matrix is equal to its inverse matrix. This can be achieved by calculating the determinant and trace of the matrix. If the determinant and trace are both zero, the matrix is irreversible, so it is not an orthogonal square matrix. Otherwise, the condition 1 can be verified by calculating the inverse matrix of the matrix and checking whether it is equal to the transposition of the matrix.
2. Next, we need to verify that all column vectors of the matrix are unit vectors and orthogonal to each other. This can be achieved by calculating the modulus length of each column vector and the dot product of any two different column vectors. If all module lengths are 1 and the dot product of any two different column vectors is zero, then the matrix satisfies condition 2.
To sum up, we can prove that a matrix is an orthogonal square by verifying whether the transposition of the matrix is equal to its inverse matrix and whether all the column vectors of the matrix are unit vectors and orthogonal.