Zero product of three matrices
0
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Let A, B, C be singular matrices such that the matrix products AB and BC are not zero. Does this imply that the product ABC is also not zero?
linear-algebra
asked Dec 18 '18 at 17:03
user10635user10635
1
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add a comment |
0
$begingroup$
Let A, B, C be singular matrices such that the matrix products AB and BC are not zero. Does this imply that the product ABC is also not zero?
linear-algebra
asked Dec 18 '18 at 17:03
user10635user10635
1
$endgroup$
add a comment |
0
0
0
$begingroup$
Let A, B, C be singular matrices such that the matrix products AB and BC are not zero. Does this imply that the product ABC is also not zero?
linear-algebra
asked Dec 18 '18 at 17:03
user10635user10635
1
$endgroup$
Let A, B, C be singular matrices such that the matrix products AB and BC are not zero. Does this imply that the product ABC is also not zero?
linear-algebra
linear-algebra
asked Dec 18 '18 at 17:03
user10635user10635
1
asked Dec 18 '18 at 17:03
user10635user10635
1
asked Dec 18 '18 at 17:03
user10635user10635
1
asked Dec 18 '18 at 17:03
user10635user10635
1
asked Dec 18 '18 at 17:03
user10635user10635
1
1
add a comment |
add a comment |
1 Answer
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The answer is no. As a counterexample, consider
$$
A = B = C = pmatrix{0&1&0\0&0&1\0&0&0}
$$
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
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votes
1
$begingroup$
The answer is no. As a counterexample, consider
$$
A = B = C = pmatrix{0&1&0\0&0&1\0&0&0}
$$
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
add a comment |
1
$begingroup$
The answer is no. As a counterexample, consider
$$
A = B = C = pmatrix{0&1&0\0&0&1\0&0&0}
$$
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
add a comment |
1
1
1
$begingroup$
The answer is no. As a counterexample, consider
$$
A = B = C = pmatrix{0&1&0\0&0&1\0&0&0}
$$
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
The answer is no. As a counterexample, consider
$$
A = B = C = pmatrix{0&1&0\0&0&1\0&0&0}
$$
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
answered Dec 18 '18 at 17:04
OmnomnomnomOmnomnomnom
129k792185
129k792185
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
add a comment |
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
I suspect that your condition holds, however, if we require that $A,B,C$ have size $2 times 2$
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:06
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
Thank you for the answer!
$endgroup$
– user10635
Dec 18 '18 at 17:08
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
As a follow-up question: given a finite set of matrices (e.g. three A,B,C), what would be the condition that guarantees that no product of any number of those matrices can be zero? Valid products would for example be ABAAC or CCB or ABC or AAACBBBA etc.
$endgroup$
– user10635
Dec 18 '18 at 17:16
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
There are certainly conditions that guarantee this outcome (e.g. if $A,B,C$ are non-singular or if $A,B,C$ commute with some additional conditions), but I don't know of any nice necessary conditions for this outcome.
$endgroup$
– Omnomnomnom
Dec 18 '18 at 17:19
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
$begingroup$
Thank you for your help!
$endgroup$
– user10635
Dec 18 '18 at 17:21
add a comment |
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