Hankel Matrix is Symmetric

Theorem

Let $\mathbf H$ be a Hankel matrix.

Then $\mathbf H$ is a symmetric matrix.

Proof

Recall the definition of symmetric matrix:

$\mathbf A$ is symmetric if and only if:

$\mathbf A = \mathbf A^\intercal$

where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.

By definition of transpose of $\mathbf A$:

$\mathbf A^\intercal_{i j}: = \mathbf A_{j i}$

Recall the definition of Hankel matrix:

A Hankel matrix is a square matrix whose antidiagonals are constant.

That is, all the elements along an antidiagonal are equal.

That is, by definition of antidiagonal:

$\forall i, j \in \closedint 1 n: \mathbf A_{i j}: = \mathbf A_{j i}$

Hence $\mathbf A^\intercal = \mathbf A$ and the result follows.

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hankel matrix