Abstract
As I have mentioned already, we may also view matrices as the representations of operators acting on the elements of an LVS. In an n-dimensional Euclidean space (denoted by \(\mathbb {R}^n\)), the natural basis is given by the ket vectors represented by the column matrices
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Notes
- 1.
I have yet to define the Hermitian conjugate of an operator, and what is meant by saying that it is Hermitian or self-adjoint (and the subtle distinction between the two). That will be done in Chap. 14, Sect. 14.2.3. For the present, take it to be that operator whose matrix representation is the Hermitian conjugate (or complex conjugate transpose) of the matrix representing the operator itself.
- 2.
Here I have used the symbol U to denote the matrix representing a general element of the group U(2), but this should not cause any confusion.
- 3.
I reiterate that we are considering finite-dimensional matrices here. In the case of infinite-dimensional matrices, it is possible for a matrix to have right eigenvectors but no left eigenvectors, or vice versa.
- 4.
Or columns. The statements that follow are valid if “row” is replaced with “column”. Recall also that the value of a determinant does not change if its rows and columns are interchanged.
- 5.
It is important to note that I am now referring to the linearly independent eigenvectors of the matrix, and not to its linearly independent rows or columns.
- 6.
As mentioned earlier, in the case of finite-dimensional matrices (which is what we are concerned with here), “self-adjoint” is the same as ‘Hermitian’. The distinction between Hermitian and self-adjoint operators in the general case will be explained in Chap. 14, Sect. 14.2.3.
- 7.
This binary operation is not to be confused with the inner product of two elements of the LVS.
- 8.
You might then ask why the generators \(J_i\) of rotations in three dimensions are Hermitian rather than antisymmetric. This has to do with real versus complex Lie algebras.
- 9.
I have used the notation \(|\psi _{i}\rangle \) and \(\langle \chi _{i}|\) for the right and left eigenvectors, so as to avoid any possible confusion with the natural basis, for which I have used the symbols \(|\phi _{i}\rangle \) and \(\langle \phi _{i}|\).
- 10.
Note that the term “spectral decomposition” is used sometimes for the representation \(S^{-1}MS = \Lambda \) of a diagonalizable matrix.
- 11.
That is, the level in each glass now becomes \(\tfrac{1}{2}(x_{0}+y_{0})\).
- 12.
Each column of M also adds up to unity. Hence M is, in fact, a doubly stochastic matrix. This last property is no longer valid in the general case, i.e., when the process is conducted with \(n > 3\) glasses, as you will see subsequently.
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Balakrishnan, V. (2020). More About Matrices. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-39680-0_12
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DOI: https://doi.org/10.1007/978-3-030-39680-0_12
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