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Let $H_n$ be the space of all $n\times n$ matrices $A = (a_{i,j})$ with entries in $\mathbb{R}$ satisfying $a_{i,j} = a_{r,s}$ whenever $i+j = r+s$ $(i, j , r , s = 1, 2, \ldots, n)$. What would be dimension of $H_n$ as a vector space over $\mathbb{R}$?

i have options for the dimension

1 - $n^2$

2- $n^2-n+1$

3 - $2n+1$

4- $2n-1$

I am finding difficulty in identifying the matrix $A$ . thanks for support

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    @Tsotsi cleared net .....2013-05-11

2 Answers 2

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HINT: Pick $n$ of moderate size and write out an example, say

$A=\pmatrix{a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}}\;.$

What are the possible values of $i+j$ for an entry $a_{ij}$? Clearly $i+j$ ranges over the set $\{2,3,\dots,8\}$. For what sets of entries is $i+j$ constant?

$A=\pmatrix{a_{11}&\color{red}{a_{12}}&\color{blue}{a_{13}}&\color{green}{a_{14}}\\ \color{red}{a_{21}}&\color{blue}{a_{22}}&\color{green}{a_{23}}&\color{purple}{a_{24}}\\ \color{blue}{a_{31}}&\color{green}{a_{32}}&\color{purple}{a_{33}}&\underline{a_{34}}\\ \color{green}{a_{41}}&\color{purple}{a_{42}}&\underline{a_{43}}&\bf a_{44}}\;.$

Now generalize to arbitrary $n$.

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    What a beautiful answer!2013-09-04
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The "lines" with $i+j$ constant can be visualized in the matrix as lines of slope $1$ (since we have our "$y$-axis" upside down): $\begin{array}{ccccc} * & \circledast & + & \oplus &\cdots\\ \circledast & + & \oplus & \times & \cdots\\ + & \oplus & \times & \otimes & \cdots\\ \oplus & \times & \otimes & \#& \cdots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}$

All entries with $*$ have the same value, because their indices add up to $2$. All entries marked with $\circledast$ have the same value, because the indices add up to $3$. All entries marked with $+$ have the same value, because the indices add up to $4$. All entries marked with $\oplus$ are equal; all entries marked with $\times$ are equal; all entries marked with $\otimes$ are equal. Etc.

The possible values of $i+j$ range from $2$ (when $i=1=j$) all the way to $2n$ (when $i=n=j$).

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    please accept my apology for that. How can i do that sir. Seeing $2n$ in your answer i got confused. I respect you. And always appreciate your answer. Pardon me.2012-05-13