3. Gram-Schmidt
Orthogonaliza-
tion Process and
Orthonormaliza-
tion
Isaac
Amornortey
Yowetu
Orthogonalization
Vrs Orthonor-
malization
Gram-Schmidt
Process
Orthogonalization Vrs
Orthonormalization
Orthogonalization
It is a process of constructing a set of orthogonal vectors
that span a particular subspace.
Suppose we have a set of linearly independent set of
vectors {v1, v2, ..., vn} ∈ V in the inner product space,
we can construct a set of orthogonal bases vectors
{u1, u2, ..., un} from the Inner Product or Euclidean space
that will be of the same subspace such that:
• the new set of vectors and the old set of vectors
must have the same linear span.
• every vector in the new set is orthogonal to every
other vector in the new set.
4. Gram-Schmidt
Orthogonaliza-
tion Process and
Orthonormaliza-
tion
Isaac
Amornortey
Yowetu
Orthogonalization
Vrs Orthonor-
malization
Gram-Schmidt
Process
Orthonormalization
It is a form of orthogonalization in which each resulting
vector is a unit vector.
Supppose {u1, u2, ..., un} as the orthogonal basis, then
w1 =
u1
||u1||
, w2 =
u2
||u2||
, ..., wn =
un
||un||
of which {w1, w2, ..., wn} be the orthonormal basis.
some Orthogonalization Algorithms
• Gram-Schmidt Process
• Householder Transformation
• Givens Rotation
Theorem
Any orthogonal set is linearly independent.
5. Gram-Schmidt
Orthogonaliza-
tion Process and
Orthonormaliza-
tion
Isaac
Amornortey
Yowetu
Orthogonalization
Vrs Orthonor-
malization
Gram-Schmidt
Process
Gram-Schmidt Process
Many Vectors
Considering a vector space V in an inner product space.
And {v1, v2, ...vn} be a basis for V, then:
u1 = v1 (1)
u2 = v2 −
< u1, v2 >
< u1, u1 >
u1 (2)
u3 = v3 −
< u1, v3 >
< u1, u1 >
u1 −
< u2, v3 >
< u2, u2 >
u2 (3)
... = · · · · · · · · · · · · · · · (4)
un = vn −
< u1, vn >
< u1, u1 >
u1−, ..., −
< un−1, vn >
< un−1, un−1 >
un−1
(5)