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In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [ 1 ] [ 2 ] It is occasionally known as adjunct matrix , [ 3 ] [ 4 ] or "adjoint", [ 5 ] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose .
The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators.Named after Bernard Helffer and Johannes Sjöstrand, this formula provides a way to calculate functions of operators without requiring the operator to have a simple or explicitly known spectrum.
In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule A x , y = x , A ∗ y , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}
Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type (Ax, y) = (x, By). Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices; Hermitian adjoint (adjoint of a linear operator) in functional analysis
The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations. The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, (), which is also sometimes called adjoint. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } with real entries reduces to the transpose of A {\displaystyle \mathbf {A} } , as the conjugate of a real number is the number itself.
Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals.
When an inline formula is long enough, it can be helpful to allow it to break across lines. Whether using LaTeX or templates, split the formula at each acceptable breakpoint into separate <math> tags or {} templates with any binary relations or operators and intermediate whitespace included at the trailing rather than leading end of a part.