## Self-Adjoint Transformations

Let’s now consider a single inner-product space and a linear transformation . Its adjoint is another linear transformation . This opens up the possibility that might be the *same* transformation as . If this happens, we say that is “self-adjoint”. It then satisfies the adjoint relation

What does this look like in terms of matrices? Since we only have one vector space we only need to pick one orthonormal basis . Then we get a matrix

That is, the matrix of a self-adjoint transformation is its own conjugate transpose. We have a special name for this sort of matrix — “Hermitian” — even though it’s exactly equivalent to self-adjointness as a linear transformation. If we’re just working over a real vector space we don’t have to bother with conjugation. In that case we just say that the matrix is symmetric.

Over a one-dimensional complex vector space, the matrix of a linear transformation is simply a single complex number . If is to be self-adjoint, we must have , and so must be a real number. In this sense, the operation of taking the conjugate transpose of a complex matrix (or the simple transpose of a real matrix) extends the idea of conjugating a complex number. Self-adjoint matrices, then, are analogous to real numbers.

[…] and Forms I Yesterday, we defined a Hamiltonian matrix to be the matrix-theoretic analogue of a self-adjoint transformation. So why should we separate out […]

Pingback by Matrices and Forms I « The Unapologetic Mathematician | June 24, 2009 |

[…] particular, this shows that if we have a symmetric form, it’s described by a self-adjoint transformation . Hermitian forms are also described by self-adjoint transformations . And […]

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[…] this all sort of makes sense. Self-adjoint transformations (symmetric or Hermitian) are analogous to the real numbers sitting inside the complex numbers. Within these, positive-definite matrices are sort of like the positive real numbers. It […]

Pingback by Positive-Definite Transformations « The Unapologetic Mathematician | July 13, 2009 |

[…] off, note that no matter what we use, the transformation in the middle is self-adjoint and positive-definite, and so the new form is symmetric and positive-definite, and thus defines […]

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[…] of the original transformation (or just the same, for a real transformation). So what about self-adjoint transformations? We’ve said that these are analogous to real numbers, and indeed their […]

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[…] All the transformations in our analogy — self-adjoint and unitary (or orthogonal), and even anti-self-adjoint (antisymmetric and […]

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[…] to denote transformations). We also have its adjoint . Then is positive-semidefinite (and thus self-adjoint and normal), and so the spectral theorem applies. There must be a unitary transformation […]

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[…] the underlying space forms a vector space itself. Indeed, such forms correspond to correspond to Hermitian matrices, which form a vector space. Anyway, rather than write the usual angle-brackets, we will write one […]

Pingback by Projecting Onto Invariants « The Unapologetic Mathematician | November 13, 2010 |

This was exactly what I needed. Plowing through Thorpe: Elementary Aspects of Differential Geometry on my own. On p. 58 he notes that the Weingarten map Lp is self-adjoint

Lp(v) dot v = Lp(w) dot v; v,w vectors in a real finite dimension vector space. Elsewhere he notes that the map is linear. So a matrix is definitely involved. This post saved me from plowing through Axler, Strang again (it’s been years) to find what such a matrix must look like.

Thanks and thanks to Google

Luysii

Comment by luysii | January 31, 2013 |