Calculate and verify the orthonormal basis vectors for the range of a full rank matrix.

Define a matrix and find the rank.

A = [1 0 1;-1 -2 0; 0 1 -1]; r = rank(A)

r = 3

自Ais a square matrix of full rank, the orthonormal basis calculated byorth(A)matches the matrixUcalculated in the singular value decomposition,[U,S] = svd(A,'econ')．This is because the singular values ofAare all nonzero.

Calculate the orthonormal basis for the range ofAusingorth．

Q = orth(A)

Q =3×3-0.1200 -0.8097 0.5744 0.9018 0.1531 0.4042 -0.4153 0.5665 0.7118

The number of columns inQis equal torank(A)．自Ais of full rank,QandAare the same size.

Verify that the basis,Q, is orthogonal and normalized within a reasonable error range.

E = norm(eye(r)-Q'*Q,'fro')

E = 1.0857e-15

The error is on the order ofeps．

Calculate and verify the orthonormal basis vectors for the range of a rank deficient matrix.

Define a singular matrix and find the rank.

A = [1 0 1; 0 1 0; 1 0 1]; r = rank(A)

r = 2

自Ais rank deficient, the orthonormal basis calculated byorth(A)matches only the firstr = 2columns of matrixUcalculated in the singular value decomposition,[U,S] = svd(A,'econ')．This is because the singular values ofAarenotall nonzero.

Calculate the orthonormal basis for the range ofAusingorth．

Q = orth(A)

Q =3×2-0.7071 -0.0000 0 1.0000 -0.7071 0.0000

自Ais rank deficient,Qcontains one fewer column thanA．