How to give counterexample for given claim
up vote
5
down vote
favorite
Suppose $A_1,dots,A_m$ be distinct $ntimes n $ real matrices such that $A_iA_j=0$ for all $ineq j$. Show that $mleq n$.
I think this true because i tried for $3times 3$ and $2times 2$ case I got only $3$ and $2$ matrices with that property.
But given that this not true .
Can any one help me to find counterexample.
And what is best approach to tackle such problem.
linear-algebra matrices examples-counterexamples
edited Nov 30 at 6:41
Robert Z
91.3k1058129
asked Nov 30 at 6:38
MathLover
3989
|
show 2 more comments
up vote
5
down vote
favorite
Suppose $A_1,dots,A_m$ be distinct $ntimes n $ real matrices such that $A_iA_j=0$ for all $ineq j$. Show that $mleq n$.
I think this true because i tried for $3times 3$ and $2times 2$ case I got only $3$ and $2$ matrices with that property.
But given that this not true .
Can any one help me to find counterexample.
And what is best approach to tackle such problem.
linear-algebra matrices examples-counterexamples
edited Nov 30 at 6:41
Robert Z
91.3k1058129
asked Nov 30 at 6:38
MathLover
3989
2
Do you require that $A_i$ is never the zero matrix?
– platty
Nov 30 at 6:42
Ohh If I take that then I get counterexample Thanks ...But if we assume all A_i nonzero then is it possible
– MathLover
Nov 30 at 6:43
@MathLover What if $A_1,dots,A_m$ are linearly independent? Are you sure that in the statement we have "distinct" matrices?
– Robert Z
Nov 30 at 6:50
Yes Statement is distant Not Linearly Indepedent .
– MathLover
Nov 30 at 6:51
@RobertZ Sir If matrices are linearly indepdent then I this above statement is true .Is I am correct?
– MathLover
Nov 30 at 6:52
|
show 2 more comments
up vote
5
down vote
favorite
up vote
5
down vote
favorite
Suppose $A_1,dots,A_m$ be distinct $ntimes n $ real matrices such that $A_iA_j=0$ for all $ineq j$. Show that $mleq n$.
I think this true because i tried for $3times 3$ and $2times 2$ case I got only $3$ and $2$ matrices with that property.
But given that this not true .
Can any one help me to find counterexample.
And what is best approach to tackle such problem.
linear-algebra matrices examples-counterexamples
edited Nov 30 at 6:41
Robert Z
91.3k1058129
asked Nov 30 at 6:38
MathLover
3989
Suppose $A_1,dots,A_m$ be distinct $ntimes n $ real matrices such that $A_iA_j=0$ for all $ineq j$. Show that $mleq n$.
I think this true because i tried for $3times 3$ and $2times 2$ case I got only $3$ and $2$ matrices with that property.
But given that this not true .
Can any one help me to find counterexample.
And what is best approach to tackle such problem.
linear-algebra matrices examples-counterexamples
linear-algebra matrices examples-counterexamples
edited Nov 30 at 6:41
Robert Z
91.3k1058129
asked Nov 30 at 6:38
MathLover
3989
edited Nov 30 at 6:41
Robert Z
91.3k1058129
asked Nov 30 at 6:38
MathLover
3989
edited Nov 30 at 6:41
Robert Z
91.3k1058129
edited Nov 30 at 6:41
Robert Z
91.3k1058129
edited Nov 30 at 6:41
Robert Z
91.3k1058129
91.3k1058129
asked Nov 30 at 6:38
MathLover
3989
asked Nov 30 at 6:38
MathLover
3989
asked Nov 30 at 6:38
MathLover
3989
3989
2
Do you require that $A_i$ is never the zero matrix?
– platty
Nov 30 at 6:42
Ohh If I take that then I get counterexample Thanks ...But if we assume all A_i nonzero then is it possible
– MathLover
Nov 30 at 6:43
@MathLover What if $A_1,dots,A_m$ are linearly independent? Are you sure that in the statement we have "distinct" matrices?
– Robert Z
Nov 30 at 6:50
Yes Statement is distant Not Linearly Indepedent .
– MathLover
Nov 30 at 6:51
@RobertZ Sir If matrices are linearly indepdent then I this above statement is true .Is I am correct?
– MathLover
Nov 30 at 6:52
|
show 2 more comments
2
Do you require that $A_i$ is never the zero matrix?
– platty
Nov 30 at 6:42
Ohh If I take that then I get counterexample Thanks ...But if we assume all A_i nonzero then is it possible
– MathLover
Nov 30 at 6:43
@MathLover What if $A_1,dots,A_m$ are linearly independent? Are you sure that in the statement we have "distinct" matrices?
– Robert Z
Nov 30 at 6:50
Yes Statement is distant Not Linearly Indepedent .
– MathLover
Nov 30 at 6:51
@RobertZ Sir If matrices are linearly indepdent then I this above statement is true .Is I am correct?
– MathLover
Nov 30 at 6:52
2
2
Do you require that $A_i$ is never the zero matrix?
– platty
Nov 30 at 6:42
Do you require that $A_i$ is never the zero matrix?
– platty
Nov 30 at 6:42
Ohh If I take that then I get counterexample Thanks ...But if we assume all A_i nonzero then is it possible
– MathLover
Nov 30 at 6:43
Ohh If I take that then I get counterexample Thanks ...But if we assume all A_i nonzero then is it possible
– MathLover
Nov 30 at 6:43
@MathLover What if $A_1,dots,A_m$ are linearly independent? Are you sure that in the statement we have "distinct" matrices?
– Robert Z
Nov 30 at 6:50
@MathLover What if $A_1,dots,A_m$ are linearly independent? Are you sure that in the statement we have "distinct" matrices?
– Robert Z
Nov 30 at 6:50
Yes Statement is distant Not Linearly Indepedent .
– MathLover
Nov 30 at 6:51
Yes Statement is distant Not Linearly Indepedent .
– MathLover
Nov 30 at 6:51
@RobertZ Sir If matrices are linearly indepdent then I this above statement is true .Is I am correct?
– MathLover
Nov 30 at 6:52
@RobertZ Sir If matrices are linearly indepdent then I this above statement is true .Is I am correct?
– MathLover
Nov 30 at 6:52
|
show 2 more comments
1 Answer
1
active
oldest
votes
up vote
7
down vote
accepted
This is false. Let $A$ be a $2times 2$ matrix such that $A^{2}=0$. The the collection ${cA:cin mathbb R}$ has this property.
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
This is false. Let $A$ be a $2times 2$ matrix such that $A^{2}=0$. The the collection ${cA:cin mathbb R}$ has this property.
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
add a comment |
up vote
7
down vote
accepted
This is false. Let $A$ be a $2times 2$ matrix such that $A^{2}=0$. The the collection ${cA:cin mathbb R}$ has this property.
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
add a comment |
up vote
7
down vote
accepted
up vote
7
down vote
accepted
This is false. Let $A$ be a $2times 2$ matrix such that $A^{2}=0$. The the collection ${cA:cin mathbb R}$ has this property.
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
This is false. Let $A$ be a $2times 2$ matrix such that $A^{2}=0$. The the collection ${cA:cin mathbb R}$ has this property.
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
answered Nov 30 at 6:44
Kavi Rama Murthy
43.9k31852
43.9k31852
add a comment |
add a comment |
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2
Do you require that $A_i$ is never the zero matrix?
– platty
Nov 30 at 6:42
Ohh If I take that then I get counterexample Thanks ...But if we assume all A_i nonzero then is it possible
– MathLover
Nov 30 at 6:43
@MathLover What if $A_1,dots,A_m$ are linearly independent? Are you sure that in the statement we have "distinct" matrices?
– Robert Z
Nov 30 at 6:50
Yes Statement is distant Not Linearly Indepedent .
– MathLover
Nov 30 at 6:51
@RobertZ Sir If matrices are linearly indepdent then I this above statement is true .Is I am correct?
– MathLover
Nov 30 at 6:52