Linear Algebra 59 | Adjoint

Adjoint matrices are introduced as the complex-number counterpart to the transpose, and they’re pinned down by how they interact with the inner product in 3n. In real vector spaces, moving a matrix from one side of an inner product to the other uses the transpose. In complex vector spaces, the same “shift” requires not only swapping rows and columns but also taking complex conjugates—turning the transpose into the adjoint (written as A*). This matters because the standard inner product in 3n is defined using complex conjugation, so the adjoint is the matrix operation that preserves the inner-product structure.

The discussion starts by recalling the inner product. For vectors in 3n, the standard inner product uses complex conjugation on the first argument: 3n’s inner product is written as X*Y (with the star indicating conjugate transpose later on). A key property from the real case is then extended: with a real matrix A, the transpose lets one “shift” A between arguments inside the inner product. The same computation in 3n shows that when indices are swapped to rewrite the expression as a matrix product, each entry of A must also be complex-conjugated. That combined operation—transpose plus entrywise conjugation—is exactly the adjoint.

A formal definition follows: for a complex mn matrix A, the adjoint A* is the nm matrix obtained by exchanging rows and columns and conjugating every entry. The adjoint is also known by other names such as the conjugate transpose or Hermitian conjugate, and some conventions use a capital H instead of a star.

An illustrative example uses a 23 matrix whose entries include i and -i. The adjoint becomes a 32 matrix: first transpose the layout, then conjugate each entry (so i becomes -i and vice versa). A special note clarifies that if A has only real entries, then A* reduces to the transpose, even though the adjoint notation remains the correct one in complex settings.

Finally, the adjoint’s spectral behavior is connected to eigenvalues. The spectrum of A* is obtained from the eigenvalues of A by complex conjugation: if bb is an eigenvalue of A, then bb̄ is an eigenvalue of A*. Geometrically, eigenvalues of A* appear as reflections of eigenvalues of A across the real axis in the complex plane. The link is justified by the fact that eigenvalues are zeros of the characteristic polynomial, and conjugating the matrix entries conjugates the polynomial’s roots accordingly. The takeaway is practical: whenever complex inner products and eigenvalue problems arise, the adjoint is the operation that correctly “moves” matrices and predicts how eigenvalues transform.