Linear Algebra 59 | Adjoint [dark version]

TL;DR

In n, the standard inner product conjugates the first argument componentwise, which changes how matrices shift across the inner product.

Briefing Cornell Notes

Briefing

Adjoint matrices are the complex-matrix counterpart of transposes, and they’re built to make the inner product work correctly in n. 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—so the transpose becomes the adjoint (also called the conjugate transpose or Hermitian conjugate). This matters because the standard inner product in n is defined using complex conjugation, and the adjoint is exactly the matrix operation that preserves the inner-product identity.

The starting point is the inner product. In n, the standard inner product takes the complex conjugate of the first argument: for vectors x and y, it is written as x* y (using the adjoint symbol on x). With that convention, the familiar transpose property from n carries over with a modification: when a complex matrix A is moved across the inner product, the appropriate replacement is A* rather than A. Concretely, the transpose alone would fail to account for conjugation; the adjoint fixes this by conjugating the matrix entries after transposing.

Formally, for a complex mn matrix A with entries a_ij, the adjoint A* is obtained by exchanging rows and columns and then complex-conjugating each entry. The result is an nm matrix. If A has only real entries, A* reduces to the transpose A, but the adjoint notation remains the correct one for complex calculations.

A quick example illustrates the rule: a 23 matrix with entries involving i becomes a 32 matrix after transposition, and every entry is conjugated—so i turns into -i in the adjoint.

The adjoint’s payoff shows up again when eigenvalues enter. The spectrum of A* is tightly linked to the spectrum of A: if is an eigenvalue of A, then its complex conjugate is an eigenvalue of A*. Geometrically, eigenvalues of A* are the mirror image of eigenvalues of A across the real axis in the complex plane. This follows from how the characteristic polynomial’s roots transform when complex conjugation is applied to the matrix entries.

Overall, the adjoint is not just a new definition—it’s the operation that makes inner products in complex vector spaces behave the way transposes do in real spaces, and it directly controls how eigenvalues transform under conjugate-transpose operations.

Cornell Notes

Adjoint matrices are the complex analogue of transposes and are defined so that inner-product identities work in n. With the standard complex inner product, the complex conjugation on the first argument forces matrix shifting across the inner product to use A* (the conjugate transpose) rather than A alone. The adjoint A* is formed by transposing A and then taking the complex conjugate of every entry; it swaps dimensions from mn to nm. Eigenvalues also transform predictably: the eigenvalues of A* are exactly the complex conjugates of the eigenvalues of A, meaning they reflect across the real axis in the complex plane.

How does the standard inner product differ between n and n, and why does that change what “moves across” a matrix?

In n (real case), the inner product is computed by multiplying corresponding components and summing, with no conjugation. In n (complex case), the standard inner product conjugates the first argument componentwise: x* y uses the complex conjugate of x’s entries. Because that conjugation is built into the inner product, shifting a matrix from one side to the other must incorporate conjugation too—so the transpose property becomes a conjugate-transpose (adjoint) property.

What is the exact definition of the adjoint matrix A* for a complex matrix A?

For a complex mn matrix A with entries a_ij, the adjoint A* is the nm matrix obtained by swapping rows and columns (transpose) and then complex-conjugating each entry. Equivalently, each entry of A* is the complex conjugate of the corresponding transposed entry of A.

Why can’t the transpose alone replace the adjoint in complex inner-product calculations?

When attempting to rewrite expressions involving A inside the complex inner product, the complex conjugation attached to the first argument forces conjugation to appear on the matrix entries as well. Transposing swaps indices, but without conjugating the entries, the resulting expression won’t match the conjugation pattern required by the inner product definition in n.

If a complex matrix A happens to have only real entries, what is A*?

If all entries of A are real, complex conjugation does nothing. In that case, A* equals the transpose A. Even so, using the adjoint notation remains the correct convention for complex-space formulas.

How do eigenvalues of A* relate to eigenvalues of A?

If is an eigenvalue of A, then its complex conjugate is an eigenvalue of A*. So the spectrum of A* is the reflection of the spectrum of A across the real axis in the complex plane. This can be checked via the characteristic polynomial: conjugating the matrix entries conjugates the roots.

Review Questions

Given a complex matrix A, what two operations (in what order) produce A*?
In the complex inner product, which argument is conjugated, and how does that affect the matrix that appears when shifting across the inner product?
If A has an eigenvalue 2+i, what eigenvalue must A* have?

Key Points

1
In n, the standard inner product conjugates the first argument componentwise, which changes how matrices shift across the inner product.
2
The adjoint A* is the correct complex analogue of the transpose for inner-product identities.
3
A* is formed by transposing A and then taking the complex conjugate of every entry; it changes dimensions from mn to nm.
4
If A has only real entries, then A* equals the transpose A.
5
Eigenvalues transform by complex conjugation: eigenvalues of A* are the complex conjugates of eigenvalues of A.
6
Geometrically, the eigenvalues of A* are reflected across the real axis compared with those of A.

Highlights

The adjoint A* is built to match the complex inner product’s conjugation on the first argument.

A* is “transpose plus entrywise conjugation,” producing an nm matrix from an mn matrix.

Eigenvalues of A* are exactly the complex conjugates of eigenvalues of A, reflecting the spectrum across the real axis.