Question

An n×nmatrix Ais said to be a Hankel matrix(named after the German mathematician Hermann Hankel, 1839–1873) if ${\mathit{a}}_{\mathbf{i}\mathbf{j}}\mathbf{=}{\mathit{a}}_{\mathbf{i}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{j}\mathbf{-}\mathbf{1}}$for all $\mathit{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}\mathbf{-}\mathbf{1}$and all$\mathit{j}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}$ meaning that Ahas constant positive sloping diagonals. For example, a 4×4 Hankel matrix is of the form.

$\mathit{A}\mathbf{=}\left[\begin{array}{cccc}a& b& c& d\\ b& c& d& e\\ c& d& e& f\\ d& e& f& g\end{array}\right]$

Show that the n×nHankel matrices form a subspace of ${\mathit{R}}^{\mathbf{n}\mathbf{\times }\mathbf{n}}$Find the dimension of this space.

Short answer

The dimension of n×n Hankel matrices is 2n-1 .

Step-by-step solution

Subspace ofRn×n

Let us denote the set of all Hankel matrices by H. Let $AB\in H$ and k be any scalar.

a)$AB\in H$

$$\begin{gathered} (A+B{)}_{ij}=A_{ij}+B_{ij} \\ =A_{i+1,j-1}+B_{i+1,j-1} \\ =(A+B{)}_{i+1,j-1} \end{gathered}$$

Thus,$AB\in H$.

b)$kA€H$

$$\begin{gathered} {\left(kA\right)}_{ij}=k{A}_{ij} \\ =k{A}_{i+1,j-1} \\ ={\left(kA\right)}_{i+1,j-1} \end{gathered}$$

Thus, $kA€H$.

Hence, H is a subspace of ${\mathrm{R}}^{\mathrm{n}\times \mathrm{n}}$.

Dimension of H.

Now, in order to find out the dimension of H. it needs to write-down an arbitrary element.

$$\begin{gathered} H=\left[\begin{array}{cc}{a}_{11}& a_{12}....a_{1n}\\ a_{21}& a_{22}....a_{2n}\\ .& ......\\ .& .....\\ a_{n1}& a_{n2}...a_{nn}\end{array}\right]:\left[a_{ij}=a_{i+j,j-1},i=1,2,...n-1,j=2,...n\right] \\ =\left[\begin{array}{cccc}{a}_{11}& a_{12}& & a_{13}........a_{1n}\\ a_{12}& & & a_{13}............a_{2n}\\ a_{13}& & & ...................a_{3n}\\ .& & & .\\ ..& & & a_{n-1,n}{a}_{n-1},n\\ a_{1n}{a}_{2n}& ..a_{n-2,n}& a_{n-1,n}& a_{nm}\end{array}\right]a_{ij}\in R \end{gathered}$$

Consider the diagram below.

Observe that, n-1 independent elements are there in the upper block and n-1 independent elements are there in the lower block and one in the diagonal.

Therefore, the total number of independent elements are

$(n-1)+1+(n-1)=2n-1$

Hence, the dimension of n×nHankel matrices is 2n-1 .