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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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Should someone signal to the common TA of preeti and srijan the fate of the exercises used in their class? –  Did May 14 '12 at 19:38
@Didier Dear sir we have no relationship with each other. We are from the same country. That may be only one link between us. Earlier i have told that i am preparing for GATE and NET (graduate aptitude tof engineering and National eligibility test) exam conducted in India. Most of the my questions are from the unsolved papers of those exam. Even i can send you links of those exam's paper. She must be preparin the same. Thia may be the sole reason of any coincoidence. To prove my genuinity i can provide you all the links of those exams.Still i beg pardon from you if you have any other thought. –  srijan May 14 '12 at 19:52
i am sorry its not tof its test hoping for your reply sir? –  srijan May 14 '12 at 20:00
srijan net peyegachho? –  Bunuelian Trick May 11 '13 at 12:57
@Tsotsi cleared net ..... –  srijan May 11 '13 at 15:01
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2 Answers

up vote 4 down vote accepted

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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@srijan: Your condition says that the entries in each of the reverse diagonals have to be constant, like $$\pmatrix{1&2&3\\2&3&4\\3&4&5}\;.$$ How many independent choices do you have to make in order to specify one of these matrices? One for each reverse diagonal. –  Brian M. Scott May 12 '12 at 6:13
Here total number of independent choices are 5? –  srijan May 12 '12 at 6:22
@srijan: That's right. In general you have to specify the value for each reverse diagonal, so one basis in the $n=3$ case consists of the matrices $$\pmatrix{1&0&0\\0&0&0\\0&0&0},\pmatrix{0&1&0\\1&0&0\\0&0&0}\;,$$ and what three other matrices? –  Brian M. Scott May 12 '12 at 6:25
$\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array} \right)$ –  srijan May 12 '12 at 6:34
$\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array} \right)$ –  srijan May 12 '12 at 6:35
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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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Dear sir what exactly we have to here to find out dimension? –  srijan May 12 '12 at 6:04
@srijan: I don't understand what you are trying to say. –  Arturo Magidin May 12 '12 at 21:07
Oh i am sorry sir . I was asking what is the dimension of given space? What exactly we have to do here? –  srijan May 12 '12 at 23:32
@srijan: Since all the entries with a given symbol have to be equal, each symbol gives you a degree of freedom. The dimension is the number of degrees of freedom. –  Arturo Magidin May 12 '12 at 23:39
Ya sir i got your point . here it should be $2n-1$. Am i right sir? –  srijan May 12 '12 at 23:42
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