# Answer to Question #16324 in Linear Algebra for Esther

Question #16324

a linear algebra question:

let A be an n*n square matrix whose columns form an orthonormal set. Compute A transpose * A

let A be an n*n square matrix whose columns form an orthonormal set. Compute A transpose * A

Expert's answer

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.

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.

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