Show y*x=0 by eigenvectors [closed]
Let $A$ be a square matrix, $lambda$ and $mu$ be two distinct eigenvalues of $A$.
If $x$ is an eigenvector corresponding to $lambda$, $y^∗$ is a nonzero row vector such that $y^∗ (A − mu I) = 0$, show that $y^∗ x = 0$.
can someone give me some direction to solve this problem?
linear-algebra
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
asked Nov 29 '18 at 1:52
CColaCCola
275
closed as off-topic by Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos Nov 29 '18 at 12:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Let $A$ be a square matrix, $lambda$ and $mu$ be two distinct eigenvalues of $A$.
If $x$ is an eigenvector corresponding to $lambda$, $y^∗$ is a nonzero row vector such that $y^∗ (A − mu I) = 0$, show that $y^∗ x = 0$.
can someone give me some direction to solve this problem?
linear-algebra
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
asked Nov 29 '18 at 1:52
CColaCCola
275
closed as off-topic by Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos Nov 29 '18 at 12:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
"Can I simply prove it by letting a square matrix?" ... letting a square matrix WHAT?
– David G. Stork
Nov 29 '18 at 1:53
add a comment |
Let $A$ be a square matrix, $lambda$ and $mu$ be two distinct eigenvalues of $A$.
If $x$ is an eigenvector corresponding to $lambda$, $y^∗$ is a nonzero row vector such that $y^∗ (A − mu I) = 0$, show that $y^∗ x = 0$.
can someone give me some direction to solve this problem?
linear-algebra
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
asked Nov 29 '18 at 1:52
CColaCCola
275
Let $A$ be a square matrix, $lambda$ and $mu$ be two distinct eigenvalues of $A$.
If $x$ is an eigenvector corresponding to $lambda$, $y^∗$ is a nonzero row vector such that $y^∗ (A − mu I) = 0$, show that $y^∗ x = 0$.
can someone give me some direction to solve this problem?
linear-algebra
linear-algebra
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
asked Nov 29 '18 at 1:52
CColaCCola
275
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
asked Nov 29 '18 at 1:52
CColaCCola
275
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
edited Nov 29 '18 at 1:59
Siong Thye Goh
99.8k1464117
99.8k1464117
asked Nov 29 '18 at 1:52
CColaCCola
275
asked Nov 29 '18 at 1:52
CColaCCola
275
asked Nov 29 '18 at 1:52
CColaCCola
275
275
closed as off-topic by Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos Nov 29 '18 at 12:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos Nov 29 '18 at 12:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Brahadeesh, José Carlos Santos, Ali Caglayan, amWhy, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
"Can I simply prove it by letting a square matrix?" ... letting a square matrix WHAT?
– David G. Stork
Nov 29 '18 at 1:53
add a comment |
"Can I simply prove it by letting a square matrix?" ... letting a square matrix WHAT?
– David G. Stork
Nov 29 '18 at 1:53
"Can I simply prove it by letting a square matrix?" ... letting a square matrix WHAT?
– David G. Stork
Nov 29 '18 at 1:53
"Can I simply prove it by letting a square matrix?" ... letting a square matrix WHAT?
– David G. Stork
Nov 29 '18 at 1:53
add a comment |
3 Answers
3
active
oldest
votes
I will assume I is the identity matrix. Since y*(A−μI)=0 then distributing yields y$^*A$=y$^*μI$ , since x is a eigen vector we have y$^*$Ax-y$^*$μIx=0$
so you need to know the fact that if you have a matrix A and a vector v such that Av=0 then A=0 or v=0.
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
add a comment |
Hint:
Use the property that
$$y^*(Ax)=(y^*A)x.$$
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
add a comment |
We may as well start with $y^*(A-mu I) = 0$ and see what happens:
$$
begin{align*}
0 &= y^*(A - mu I)\
&= y^*(A - mu I)x\
&= y^*(Ax - mu x)\
&= ldots
end{align*}
$$
... can you take it from here?
spoiler alert
$$0 = y^*(Ax - mu x) = y^*(lambda x - mu x) = y^*x(lambda - mu)$$ and as $lambda neq mu$ we must have $y^*x = 0$.
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
I will assume I is the identity matrix. Since y*(A−μI)=0 then distributing yields y$^*A$=y$^*μI$ , since x is a eigen vector we have y$^*$Ax-y$^*$μIx=0$
so you need to know the fact that if you have a matrix A and a vector v such that Av=0 then A=0 or v=0.
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
add a comment |
I will assume I is the identity matrix. Since y*(A−μI)=0 then distributing yields y$^*A$=y$^*μI$ , since x is a eigen vector we have y$^*$Ax-y$^*$μIx=0$
so you need to know the fact that if you have a matrix A and a vector v such that Av=0 then A=0 or v=0.
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
add a comment |
I will assume I is the identity matrix. Since y*(A−μI)=0 then distributing yields y$^*A$=y$^*μI$ , since x is a eigen vector we have y$^*$Ax-y$^*$μIx=0$
so you need to know the fact that if you have a matrix A and a vector v such that Av=0 then A=0 or v=0.
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
I will assume I is the identity matrix. Since y*(A−μI)=0 then distributing yields y$^*A$=y$^*μI$ , since x is a eigen vector we have y$^*$Ax-y$^*$μIx=0$
so you need to know the fact that if you have a matrix A and a vector v such that Av=0 then A=0 or v=0.
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
answered Nov 29 '18 at 2:10
maths researchermaths researcher
508
508
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
add a comment |
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
ok thanks! i will try to approach in this way
– CCola
Nov 29 '18 at 4:11
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
No problem, if my answer was helpful I would appreciate it if you click on the check mark
– maths researcher
Nov 29 '18 at 4:15
add a comment |
Hint:
Use the property that
$$y^*(Ax)=(y^*A)x.$$
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
add a comment |
Hint:
Use the property that
$$y^*(Ax)=(y^*A)x.$$
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
add a comment |
Hint:
Use the property that
$$y^*(Ax)=(y^*A)x.$$
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
Hint:
Use the property that
$$y^*(Ax)=(y^*A)x.$$
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
answered Nov 29 '18 at 1:57
Siong Thye GohSiong Thye Goh
99.8k1464117
99.8k1464117
add a comment |
add a comment |
We may as well start with $y^*(A-mu I) = 0$ and see what happens:
$$
begin{align*}
0 &= y^*(A - mu I)\
&= y^*(A - mu I)x\
&= y^*(Ax - mu x)\
&= ldots
end{align*}
$$
... can you take it from here?
spoiler alert
$$0 = y^*(Ax - mu x) = y^*(lambda x - mu x) = y^*x(lambda - mu)$$ and as $lambda neq mu$ we must have $y^*x = 0$.
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
add a comment |
We may as well start with $y^*(A-mu I) = 0$ and see what happens:
$$
begin{align*}
0 &= y^*(A - mu I)\
&= y^*(A - mu I)x\
&= y^*(Ax - mu x)\
&= ldots
end{align*}
$$
... can you take it from here?
spoiler alert
$$0 = y^*(Ax - mu x) = y^*(lambda x - mu x) = y^*x(lambda - mu)$$ and as $lambda neq mu$ we must have $y^*x = 0$.
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
add a comment |
We may as well start with $y^*(A-mu I) = 0$ and see what happens:
$$
begin{align*}
0 &= y^*(A - mu I)\
&= y^*(A - mu I)x\
&= y^*(Ax - mu x)\
&= ldots
end{align*}
$$
... can you take it from here?
spoiler alert
$$0 = y^*(Ax - mu x) = y^*(lambda x - mu x) = y^*x(lambda - mu)$$ and as $lambda neq mu$ we must have $y^*x = 0$.
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
We may as well start with $y^*(A-mu I) = 0$ and see what happens:
$$
begin{align*}
0 &= y^*(A - mu I)\
&= y^*(A - mu I)x\
&= y^*(Ax - mu x)\
&= ldots
end{align*}
$$
... can you take it from here?
spoiler alert
$$0 = y^*(Ax - mu x) = y^*(lambda x - mu x) = y^*x(lambda - mu)$$ and as $lambda neq mu$ we must have $y^*x = 0$.
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
edited Nov 29 '18 at 13:16
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
answered Nov 29 '18 at 2:00
NealNeal
23.5k23785
23.5k23785
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
add a comment |
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
so we get y*=0, x=0 or (A-μ)=0?
– CCola
Nov 29 '18 at 5:37
add a comment |
"Can I simply prove it by letting a square matrix?" ... letting a square matrix WHAT?
– David G. Stork
Nov 29 '18 at 1:53