Hankel Matrix is Symmetric
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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