a linear algebra question:
let A be an n*n square matrix whose columns form an orthonormal set. Compute A transpose * A
1
Expert's answer
2012-10-15T11:42:35-0400
Write A = [a_1 a_2 ... a_n] where a_i is a vector column. Orthonormality of these vectors means that <a_i, a_j> = 0 for i<>j and <a_i, a_j> = 1 for i=j
Notice that transpose (A) is a matrix
transpose(A) = [ a'_1 ] [ a'_2 ] [ .... ] [ a'_n ] where a'_i is a transposed to a_i
Hence the (i,j)-th element b_{i,j} of the matrix transpose(A) * A is the scalar product <a_i, a_j>
Hence b_{i,j} = 0 for i<>j and b_{i,j} = 1 for i=j
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