Inner product- Let F be the field of real or Complex numbers and V vector space over F. An inner product on V is a fuction which assigns to each ordered pair of vectors α ,β ∈ V a scalar (α/β) ∈ F, defined as
For α ,β , γ ∈ V, and all scalars C
(a) (α + β/γ) = (α/γ) + (β/γ)
(b) (cα/β) = c(α/β)
(c) (β/α) = (α̅/̅β̅) (a complex conjugate)
(d) (α/α) > 0 if α ≠ 0.
Inner product spaces- An inner product space is a real or complex space with a specified inner product on that space.
Norm- If ||α||2 = (α/α), then ||α|| is called norm of (α/α).
Euclidean space- A finite dimensional real inner product space.
Unitary space- A complex inner product space.
Orthogonal set- Let α and β be vectors in an inner product space V. Then α is orthogonal to β if (α/β) = 0.
Orthonormal set- The orthogonal set S, with property ||α|| = 1, ∀α∈S.
Orthogonal complement- Let V be a inner product space and S⊂V. The othogonal complement of S is the set S⊥ of all vectors in V which are orthogonal to every vector in S.
T preserves inner product- Let V and W be inner product spaces over the same field and T be a linear transformation from V into W. We say T preserves inner products if
(Tα/Tβ) = (α/β) for all α,β ∈ V.
Isomorphism- An isomorphism of V onto W is a vector space isomorphism T of V onto W, whick also preserves inner product,
i.e., (Tα/Tβ) = (α/β) for all α,β ∈ V.
Unitary operator- A unitary operator on an inner product space is an isomorphism of the space onto itself.
Unitary matrix- A complex square matrix A is unitary if A*A = I.
Orthogonal matrix- A real or complex square matrix A is orthogonal if ATA = I.
Unitary equivalent- If A and B are complex n×n matrices then B is unitarily equivalent to A if there exist unitary matrix P of order n×n : B = P-1AP.
Orthogonally equivalent- If A and B are complex matrices then B is orthogonally equivalent to A if there exist orthogonal matrix P of order n×n : B = P-1AP.
Normal Operators:
Let V ne a finite dimensional inner product space and T is linear operator on V. Then T is normal if TT* = T*T.
Normal matrix- A complex n×n matrix A is called normal if AA* = A*A.
For α ,β , γ ∈ V, and all scalars C
(a) (α + β/γ) = (α/γ) + (β/γ)
(b) (cα/β) = c(α/β)
(c) (β/α) = (α̅/̅β̅) (a complex conjugate)
(d) (α/α) > 0 if α ≠ 0.
Inner product spaces- An inner product space is a real or complex space with a specified inner product on that space.
Norm- If ||α||2 = (α/α), then ||α|| is called norm of (α/α).
Euclidean space- A finite dimensional real inner product space.
Unitary space- A complex inner product space.
Orthogonal set- Let α and β be vectors in an inner product space V. Then α is orthogonal to β if (α/β) = 0.
Orthonormal set- The orthogonal set S, with property ||α|| = 1, ∀α∈S.
Orthogonal complement- Let V be a inner product space and S⊂V. The othogonal complement of S is the set S⊥ of all vectors in V which are orthogonal to every vector in S.
T preserves inner product- Let V and W be inner product spaces over the same field and T be a linear transformation from V into W. We say T preserves inner products if
(Tα/Tβ) = (α/β) for all α,β ∈ V.
Isomorphism- An isomorphism of V onto W is a vector space isomorphism T of V onto W, whick also preserves inner product,
i.e., (Tα/Tβ) = (α/β) for all α,β ∈ V.
Unitary operator- A unitary operator on an inner product space is an isomorphism of the space onto itself.
Unitary matrix- A complex square matrix A is unitary if A*A = I.
Orthogonal matrix- A real or complex square matrix A is orthogonal if ATA = I.
Unitary equivalent- If A and B are complex n×n matrices then B is unitarily equivalent to A if there exist unitary matrix P of order n×n : B = P-1AP.
Orthogonally equivalent- If A and B are complex matrices then B is orthogonally equivalent to A if there exist orthogonal matrix P of order n×n : B = P-1AP.
Normal Operators:
Let V ne a finite dimensional inner product space and T is linear operator on V. Then T is normal if TT* = T*T.
Normal matrix- A complex n×n matrix A is called normal if AA* = A*A.
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