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Section 11.2 Terminology and notation

complex linear equation

a linear equation where the coefficients are complex numbers and the variables are allowed to take on complex values

complex matrix

a matrix whose entries are complex numbers

Remark 11.2.1.

Remember that a real number is just a special kind of complex number. Therefore,

  • every linear equation with real coefficients can be considered as a complex linear equation by allowing the variables involved to take on complex values; and
  • every matrix whose entries are real numbers can also be considered as a complex matrix.
complex conjugate of a matrix \(A\)

the matrix \(\cconj{A}\) obtained by replacing each entry by its complex conjugate

(complex) adjoint of a matrix \(A\)

the matrix \(\adjoint{A}\) obtained by taking the transpose of the conjugate of \(A\text{,}\) i.e. \(\adjoint{A} = \utrans{\cconj{A}}\)

self-adjoint matrix

a complex matrix that is equal to its own complex adjoint; also referred to as a Hermitian matrix

Warning 11.2.2.

Do not confuse the complex adjoint of a matrix with the classical adjoint studied in Chapter 10.