00:01
We are asked to prove theorem about orthogonal matrices.
00:08
So let a be a real n -by -n -n -in -verbal matrix.
00:14
We want to show that a is an orthogonal matrix, if and only if, the row or column vectors of a forming an orthogonal set of vectors.
00:33
So we have that a is going to be an orthogonal matrix, if and only if, by the definition.
00:45
Of orthogonal, a transpose is equal to a inverse.
00:54
And this is true, if and only if, a transpose a is equal to the identity matrix.
01:08
Since a is a real n by matrix, let a be, the matrix whose column vectors are c1 through cn.
01:28
Then we have that the ith row, of the matrix a transpose is equal to c sub i column c sub i transposed so we have the i j entry of a transpose a is going to be the inner product of the row i which is the i which is c i which is c i transpose of a transpose of a transpose interproduct with the j -th column of a, which is cj.
02:37
And so this implies that c -i transpose cj, is going to equal to, because the ijth entry is the 8th entry of i .n.
02:54
As well, is it going to be equal to the chronicter delta ij, which we know is going to be 0 when i is not equal to j...