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For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and $T^{*} T = I$ are equivalent

Statement

For an endomorphism $T$ of a finite-dimensional real or complex inner product space $V$ , the following are equivalent:

$T$ preserves norms.
$T$ preserves inner products.
$T$ sends every orthonormal basis to an orthonormal basis.
$T$ sends some orthonormal basis to an orthonormal basis.
$T^{*} T = I$ .

Whenever these conditions hold, $T$ is invertible and $T^{- 1} = T^{*}$ , so also $T T^{*} = I$ . The zero-dimensional case is included.

Facts & Assumptions

Given: An endomorphism $T$ of a finite-dimensional inner product space $V$ .

[L1]

Real and complex polarisation identities recover the inner product from the norm (Pythagoras, the parallelogram identity, and the real and complex polarisation identities).

[L2]

Every finite-dimensional inner product space has an orthonormal basis (Every finite-dimensional real or complex inner product space has an orthonormal basis).

[L3]

An adjoint is characterised by $⟨ T v, w ⟩ = ⟨ v, T^{*} w ⟩$ , and adjoint algebra gives $(S T)^{*} = T^{*} S^{*}$ and $T^{* *} = T$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ ).

[L4]

Operator determinants are multiplicative, and a finite-dimensional endomorphism is invertible exactly when its determinant is nonzero (For endomorphisms $S$ and $T$ of one finite-dimensional vector space, $\det (S T) = \det (S) \det (T)$ , A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero).

[L5]

A linear isometry is a linear map preserving every vector norm (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces).

[L6]

If $⟨ z, w ⟩ = 0$ for every $w$ in an inner product space, then $z = 0$ (Inner products separate vectors, and the induced norm is homogeneous: $∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$ ).

[L7]

Every finite orthogonal list of nonzero vectors is linearly independent (Every finite orthogonal list of nonzero vectors is linearly independent).

[L8]

A subspace of a finite-dimensional space has the same dimension as the ambient space exactly when it is the whole space (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

Proof

technique · equivalence

1.1

If $T$ preserves norms, substitute $T u, T v$ into the appropriate real or complex polarisation identity [L1]. Every norm term is unchanged, so $⟨ T u, T v ⟩ = ⟨ u, v ⟩$ . Thus (1) implies (2).

L1L5

1.2

If $T$ preserves inner products, it sends every orthonormal basis to an orthonormal list. By [L7] this list is independent; its span therefore has dimension $\dim V$ , so [L8] makes it all of $V$ . Thus (2) implies (3), while (3) implies (4) by the existence in [L2].

L2L7L8

1.3

Suppose an orthonormal basis $(e_{i})$ has orthonormal image $(T e_{i})$ . Expanding arbitrary $u, v$ in $(e_{i})$ shows directly that $⟨ T u, T v ⟩ = ⟨ u, v ⟩$ . Hence (4) implies (2), and setting $u = v$ shows (2) implies (1).

L2algebra

1.4

By the defining adjoint identity [L3], (2) is equivalent to $⟨ u, (T^{*} T - I) v ⟩ = 0$ for all $u, v$ . Conjugate symmetry and nondegeneracy [L6] make this equivalent to $(T^{*} T - I) v = 0$ for every $v$ , hence to $T^{*} T = I$ . Thus (2) and (5) are equivalent.

L3L6algebra

1.5

Under (5), multiplicativity in [L4] gives $\det (T^{*}) \det (T) = 1$ , so $\det (T) \neq 0$ . Hence $T$ is invertible and $T^{*} T = I$ gives $T^{- 1} = T^{*}$ . Consequently $T T^{*} = I$ .

2.1

All implications remain valid for the empty orthonormal basis of $V = 0$ , where the identity endomorphism is the unique map.

L2∎

Depends on

Used by

Orthogonal and unitary operators form groups, and their determinants have modulus one Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 128 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., results 7.45, 7.49, and 7.53 (standard reference, not scraped)

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T∗T=IT∗T=I are equivalent