Answer to Question #16324 in Linear Algebra for Esther

Question

a linear algebra question:
let A be an n*n square matrix whose columns form an orthonormal set. Compute A transpose * A

Answer 1

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

Therefore

transpose(A) * A
is the unit n*n matrix.