Hermitian Forms

The standard inner product of $\mathbf{C}^n$ is given by

\[\langle x, y \rangle = \sum_{i=1}^n \overline{x_i} y_i\]

which is not a bilinear form since it is not linear in the first variable. However, it is linear up to complex conjugate in which we can develop similar properties as that of symmetric bilinear forms.

Definition. Let $V$ and $W$ be vector spaces over $\mathbf{C}$. Then a sesquilinear is a function $\phi: V \times W \to \mathbf{C}$ such that

\[\begin{align*} \phi(\lambda_1 v_1 + \lambda_2 v_2, w) &= \overline{\lambda_1} \phi(v_1, w) + \overline{\lambda_2} \phi(v_2, w) \\ \phi(v, \mu_1 w_1 + \mu_2 w_2) &= \mu_1 \phi(v, w_1) + \mu_2 \phi(v, w_2) \end{align*}\]

for all $\lambda_1, \lambda_2, \mu_1, \mu_2 \in \mathbf{C}$, $v, v_1, v_2 \in V$ and $w, w_1, w_2 \in W$.

Definition. A sesquilinear form $\phi: V \times V \to \mathbf{C}$ is Hermitian if $\phi(x, y) = \overline{\phi(y, x)}$ for all $x, y \in V$.

Proposition. If $\phi$ is a Hermitian form on $V$ then $\phi(v, v) \in \mathbf{R}$ for all $v \in V$ and $\phi(\lambda v, \lambda v) = \vert \lambda \vert^2 \phi(v, v)$.

Proof.

By definition, we have $\phi(v, v) = \overline{\phi(v, v)}$ so $\phi(v, v) \in \mathbf{R}$ and $\phi(\lambda v, \lambda v) = \overline{\lambda} \lambda \phi(v, v) = \vert \lambda \vert^2 \phi(v, v)$.

Therefore, it is also meaning to speak of positive/negative (semi-)definite Hermitian forms.

Proposition. [Polarization Identity] A Hermitian form $\phi$ on a vector space $V$ over $\mathbf{C}$ is determined by the function $\psi: V \to \mathbf{R}, v \mapsto \phi(v, v)$.

Proof.

The proof is nearly identical to that of quadratic form with the following equality

\[\phi(x, y) = {1 \over 4} (\psi(x + y) - i \psi(x + iy) - \psi(x - y) + i \psi(x - iy))\]

Matrix Representation

Definition. Suppose that $V$ has a basis $(v_1, …, v_m)$ and $W$ has a basis $(w_1, …, w_n)$. Then the matrix $A$ representing $\phi$ with respect to these bases is defined by $A_{ij} = \phi(v_i, w_j)$.

Similarity, let $x = \sum \lambda_i v_i$ and $y = \sum \mu_j w_j$, we have

\[\phi(x, y) = \phi \left(\sum_i \lambda_i v_i, \sum_j \mu_j w_j \right) = \sum_{i,j} \overline{\lambda_i} \phi(v_i, w_j) \mu_j = \overline{x}^\intercal A y\]

Proposition. Suppose that $\phi: V \times V \to \mathbf{C}$ is a sesquilinear form on a $\mathbf{C}$-vector space with basis $(v_1, …, v_n)$. Then $\phi$ is Hermitian iff the matrix representing $\phi$ with respect to this basis satisfies $A = A^\dagger = \overline{A}^\intercal$, and the matrix is said to be Hermitian.

Proof.

($\Rightarrow$) If $\phi$ is Hermitian, then

\[A_{ij} = \phi(v_i, v_j) = \overline{\phi(v_j, v_i)} = \overline{A_{ji}}\]

($\Leftarrow$) If $A$ is Hermitian, then

\[\phi(x, y) = \sum_{i, j} \overline{x_i} A_{ij} y_j = \sum_{j, i} \overline{\overline{y_j} A_{ji} x_i} = \overline{\phi(y, x)}\]

Proposition. [Change of Basis] Suppose that $\phi$ is a Hermitian form on a finite dimensional complex vector space $V$ and $(e_i)$ and $(f_i)$ are bases for $V$ such that $f_i = \sum P_{ki} e_k$. If $A$ is the matrix representing $\phi$ with respect to $(e_i)$ and $B$ is that with respect to $(f_i)$ then

\[B = P^\dagger A P\]

Proof.

\[B_{ij} = \phi(f_i, f_j) = \phi \left( \sum_k P_{ki} e_k, \sum_l P_{lj} e_l \right) = \sum_{k, l} \overline{P_{ki}} A_{kl} P_{lj} = (P^\dagger A P)_{ij}\]

Proposition. [Sylvester’s Law of Inertia] Let $\phi$ be a Hermitian form on a finite dimensional $\mathbf{C}$-vector space $V$. Then there are unique integers $p, q$ such that $V$ has basis $(v_1, …, v_n)$ such that the matrix representing $\phi$ with respect to it is of the form

\[\begin{pmatrix} I_p & 0 & 0 \\ 0 & -I_q & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}\]

Proof.

The proof is nearly identical to the real case. By choosing some $v_1$ such that $\phi(v_1, v_1) \not= 0$ then we can normalize it by $v_1 / \vert \phi(v_1, v_1) \vert^{1/2}$ and then followed by induction on $U = \ker \phi(v_1, -)$. We also have $p + q = r(\phi)$ and $p$ is the maximum dimension of positive definite subspace as in real symmetric case.

Reference