Orthogonal Matrices are Isometries

We define an n by n matrix, A, to be an orthogonal matrix if A^T is A^{-1}, that is, the transpose of the matrix is its inverse. When this is the case, the columns of the matrix are an orthonormal basis of R^n. These matrices preserve the length of any input vector and we therefore call them isometries.

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[volkandemir]

Sir, how can U dot V can be equal to u transpose times v. left side is a scalar and right side is a 1x1 matrix. We can multiply left side by a 2x2 matrix but we cant multiply right side by a 2x2 matrix. So, we can easily see that 1x1 matrices are not scalars. Aren't we?