This is a beautiful problem from ISI MStat PSB 2015 Problem 2. We provide detailed solution with prerequisite mentioned explicitly.
For any matrix
consider the following three proper-
ties:
(a) Show that is a vector space for any
.
(b) Find the dimension of , when n = 2 and n = 3.
(a) To show that is a vector space for any
So, here if we can show that is a subspace of the vector space of
real matrices with usual matrix addition and scalar multiplication then we are done!
Let's try to show this ,
Putting for all i,j then
satisfies all the properties (1),(2) & (3) .
So,
Shall show that (i) for all
,
and
(ii) for all
for all
{
}-{0} ,
For (i) Take any
Let , D= and if
then
Now we will see whether D satisfies all the three properties (1),(2) and (3)
when
and
Hence as A and B are upper triangular matrix , D is also an upper triangular matrix .
So it satisfies property (1)
Again , for all
and
for all
,
then for all
as
Hence it satisfies property (2) .
Now we have for all
and
for all
,then
for all
as
Hence it satisfies the properties (3)
For (ii) Take any
take any {
}-{0}
Let, and if
then
Then , when
Hence as A is an upper triangular matrix , K is also an upper triangular matrix .
So it satisfies property (1)
Again , for all
then
for all
as
Hence it satisfies property (2) .
Now we have for all
,then
for all
as
Hence it satisfies the properties (3)
So, is closed under vector addition and scalar multiplication.
Therefore , is a subspace of the vector space of
real matrices with usual matrix addition and scalar multiplication . Hence we are done !
(b) n=2 ,
then ,
by property (1) ,
---(I) by property (2) and
---(II) by property (3) .
Now solving (I) and (II) we get
Giving , = {
} hence
n=3
then ,
by property (1) ,
---(I) by property (2) and
---(II) by property (3) .
Now solving (I) and (II) we get
(say) then ,
,
Giving , = {t
} ,
.
Hence ,
This is a beautiful problem from ISI MStat PSB 2015 Problem 2. We provide detailed solution with prerequisite mentioned explicitly.
For any matrix
consider the following three proper-
ties:
(a) Show that is a vector space for any
.
(b) Find the dimension of , when n = 2 and n = 3.
(a) To show that is a vector space for any
So, here if we can show that is a subspace of the vector space of
real matrices with usual matrix addition and scalar multiplication then we are done!
Let's try to show this ,
Putting for all i,j then
satisfies all the properties (1),(2) & (3) .
So,
Shall show that (i) for all
,
and
(ii) for all
for all
{
}-{0} ,
For (i) Take any
Let , D= and if
then
Now we will see whether D satisfies all the three properties (1),(2) and (3)
when
and
Hence as A and B are upper triangular matrix , D is also an upper triangular matrix .
So it satisfies property (1)
Again , for all
and
for all
,
then for all
as
Hence it satisfies property (2) .
Now we have for all
and
for all
,then
for all
as
Hence it satisfies the properties (3)
For (ii) Take any
take any {
}-{0}
Let, and if
then
Then , when
Hence as A is an upper triangular matrix , K is also an upper triangular matrix .
So it satisfies property (1)
Again , for all
then
for all
as
Hence it satisfies property (2) .
Now we have for all
,then
for all
as
Hence it satisfies the properties (3)
So, is closed under vector addition and scalar multiplication.
Therefore , is a subspace of the vector space of
real matrices with usual matrix addition and scalar multiplication . Hence we are done !
(b) n=2 ,
then ,
by property (1) ,
---(I) by property (2) and
---(II) by property (3) .
Now solving (I) and (II) we get
Giving , = {
} hence
n=3
then ,
by property (1) ,
---(I) by property (2) and
---(II) by property (3) .
Now solving (I) and (II) we get
(say) then ,
,
Giving , = {t
} ,
.
Hence ,