Adjoints satisfy $(S+T{)}^{\ast }=S^{\ast }+T^{\ast }$, $(\lambda T{)}^{\ast }=\bar{\lambda }{T}^{\ast }$, $(ST{)}^{\ast }=T^{\ast }{S}^{\ast }$, and $T^{\ast \ast }=T$

Depends on

• The adjoint $T^*:W\to V$ is characterised by $\langle Tv,w\rangle_W=\langle v,T^*w\rangle_V$
• Every linear map between finite-dimensional inner product spaces has a unique adjoint
• Inner products separate vectors, and the induced norm is homogeneous: $\lVert\lambda v\rVert=|\lambda|\lVert v\rVert$

Used by

• An endomorphism and its adjoint have the same singular values Corollary
• Distance corrects unlocated errors and known erasures Corollary
• Orthogonal and unitary operators form groups, and their determinants have modulus one Corollary
• Finite adiabatic path model Definition
• Finite block encoding Definition
• Inverse Unitary Access Definition
• Matrix multiplicative-weights game and update Definition
• Normalized output law for finite quantum query branches Definition
• The singular values of a linear map as the eigenvalues of the positive square root of T^*T Definition
• A degenerate code with a non-diagonal Knill–Laflamme matrix Example
• Orthogonal error spaces and a recovery branch Example
• Qubit dephasing channel in Kraus and Choi forms Example
• FALSE: On a complex inner product space, (λ T)^*=λ T^* for every scalar False statement
• A unitary error-basis change gives orthogonal error ranges Lemma
• Complete positivity implies Choi positivity Lemma
• Detectable operators form a subspace and quantum distance is well defined Lemma
• Operator perturbation bound for QSVT Lemma
• Orthogonal error restrictions determine syndrome projections Lemma
• Preserving the logical marginal forces a gauge-only channel Lemma
• PUE singular-space invariant decomposition Lemma
• Singular-value transform is independent of SVD choices Lemma
• Spectrum of the lazy Johnson walk Lemma
• The finite positive square root preserves operator order Lemma
• The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order Lemma
• Trace of a product of Hermitian matrices is real Lemma
• For normal endomorphisms, the spectral functional calculus respects sums, products, adjoints, and composition of scalar functions Proposition
• Positive-definite overlap whitening preserves a generalized Ritz pencil Proposition
• Trace Holder Inequality Proposition
• Unitary conjugation is a channel Proposition
• Unitary conjugation is invariant under a global phase Proposition
• A finite Kraus sum is completely positive Theorem
• Adjoint block encoding from inverse access Theorem
• An explicit syndrome recovery corrects a Knill–Laflamme error family Theorem
• An isometric quantum encoding gives an orthogonal code projector Theorem
• An isometry followed by partial trace is a channel Theorem
• Every finite instrument induces a unique POVM Theorem
• Every finite-dimensional channel has a finite isometric dilation Theorem
• Exact recovery implies the Knill–Laflamme condition Theorem
• Exact row-preparation access count for one and two walk steps Theorem
• Finite-dimensional channels compose and tensor Theorem

…and 6 more results.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sheldon Axler, Linear Algebra Done Right, 4th ed., result 7.5 (standard reference, not scraped)
• Sergei Treil, Linear Algebra Done Wrong, §5.5.1, useful formulas for adjoints (standard reference, not scraped)