Linear Algebra · Visual Note

The Adjoint A*

Yehor Korotenko

The adjoint lets you "move a matrix to the other side" of an inner product, without changing the result.

The core identity

⟨ x, Ay ⟩ = ⟨ A*x, y ⟩

Both sides give the same scalar. On the left, A acts on y. On the right, A* acts on x instead. The adjoint is defined by this property — it's whatever makes this equality hold.

How to think about it

The inner product ⟨x, y⟩ measures "how much x and y point in the same direction." It returns a single real number (for Hermitian matrices).

When you write ⟨x, Ay⟩, you first transform y with A, then measure against x.

Think: "after A stretches y, how much does it align with x?"

The adjoint A* is the unique operator such that you get the same number by transforming x instead of y.

⟨x, Ay⟩ = ⟨A*x, y⟩ — always, for all x and y.

In coordinates: A* is obtained by transposing A and conjugating its entries. For real matrices, A* = A^T.

(A*)ᵢⱼ = conj(Aⱼᵢ)

When A* = A, the matrix is Hermitian: it treats both "slots" of the inner product identically. This forces eigenvalues to be real.

Interactive demo — 2D real case

Drag the x or y vectors. Watch both inner products stay equal.

A₁₁ row 1, col 1

1.50

A₁₂ row 1, col 2

0.50

A₂₁ row 2, col 1

-0.50

A₂₂ row 2, col 2

0.80

⟨x, Ay⟩

—

left side

⟨A*x, y⟩

—

right side — always equal ✓

x (drag me)

y (drag me)

A*x

projection (inner product)

In the Hermitian mode, notice that Ay and A*x are "mirror images" of each other — because A* = A, the transformation is the same on both sides. This geometric symmetry is what it means for a matrix to be self-adjoint.