Suppose that upsilon1 upsilonn is an orthonormal basis for

Suppose that {upsilon_1, ..., upsilon_n} is an orthonormal basis for R^n with the usual dot product. Construct a matrix A = [upsilon_1|upsilon_2|...|upsilon_n]. Compute A^T A and (you must justify why your answer is what it is, and you cannot just use an example). What does your computation tell you about A^T? Do you remember what the name is for a matrix like A, and does it now make sense?

Solution

Suppose that { v₁ , v₂ , v_3,…, v_n } is an orthonormal basis for Rⁿ with dot product. Let A be a matrix with columns v₁ , v₂ , …, v_n . Also let v_i = ( a_1i ,a_2i , a_3i ,…, a_ni)^T. Then the columns of A^T are u₁ , u₂ , u₃ , …, u_n where u_i = ( a_i1, a_i2, a_i3, …, a_in)^T 1 I n. Then A^T A has columns c₁ , c₂ , c₃ ,…, c_n where c_i = (c_1i,c_2i, c_3i,…, c_ni)^T. Further, c_ii = a_1i²+a_2i²+ a_3i² + … +a_ni² (1 I n) = 1 as each v_i has magnitude1. Further c_ij ( i uj and 1 I n) = a_1i a_1j +a_2i a_2j +a_3i a_3j +…+a_ni a_nj = v_i . v_j = 0. Thus A^T A = I_n. Similarly AA^T = I_n. This means that A is an orthogonal matrix . We know that an orthogonal matrix is a square matrix with real entries whose columns are  orthonormal vectors. A conforms to this definition.