Is every non singular matrix diagonalizable? [closed]
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We see that any matrix is diagonalizable if the sum of geometric multiplicity of all eigenvalues is equal to the sum of algebraic multiplicity of all eigenvalues. We calculate geometric multiplicity by checking no. of linearly independent column vectors.
We know that columns of a non singular matrix $A$ are linearly independent. Is A diagonalizable?
linear-algebra
edited Aug 26 at 19:47
Brahadeesh
4,11331550
asked Aug 26 at 19:38
Debprasad Kundu
2
closed as off-topic by Shaun, Morgan Rodgers, TheGeekGreek, Arnaud D., Nex Aug 27 at 19:17
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." – Shaun, Morgan Rodgers, TheGeekGreek, Nex
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up vote
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down vote
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We see that any matrix is diagonalizable if the sum of geometric multiplicity of all eigenvalues is equal to the sum of algebraic multiplicity of all eigenvalues. We calculate geometric multiplicity by checking no. of linearly independent column vectors.
We know that columns of a non singular matrix $A$ are linearly independent. Is A diagonalizable?
linear-algebra
edited Aug 26 at 19:47
Brahadeesh
4,11331550
asked Aug 26 at 19:38
Debprasad Kundu
2
closed as off-topic by Shaun, Morgan Rodgers, TheGeekGreek, Arnaud D., Nex Aug 27 at 19:17
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." – Shaun, Morgan Rodgers, TheGeekGreek, Nex
What about $beginbmatrix1 & 1\ 0 & 1 endbmatrix$?
– Calculon
Aug 26 at 19:41
1
You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Aug 26 at 19:42
Take a look at en.wikipedia.org/wiki/Diagonalizable_matrix It is a good idea to search for answers to questions like this before asking. This way you will understand why all non-singular matrices are not diagonalizable instead of being presented with some useless counter-example.
– John Douma
Aug 26 at 19:48
add a comment |Â
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
We see that any matrix is diagonalizable if the sum of geometric multiplicity of all eigenvalues is equal to the sum of algebraic multiplicity of all eigenvalues. We calculate geometric multiplicity by checking no. of linearly independent column vectors.
We know that columns of a non singular matrix $A$ are linearly independent. Is A diagonalizable?
linear-algebra
edited Aug 26 at 19:47
Brahadeesh
4,11331550
asked Aug 26 at 19:38
Debprasad Kundu
2
We see that any matrix is diagonalizable if the sum of geometric multiplicity of all eigenvalues is equal to the sum of algebraic multiplicity of all eigenvalues. We calculate geometric multiplicity by checking no. of linearly independent column vectors.
We know that columns of a non singular matrix $A$ are linearly independent. Is A diagonalizable?
linear-algebra
edited Aug 26 at 19:47
Brahadeesh
4,11331550
asked Aug 26 at 19:38
Debprasad Kundu
2
edited Aug 26 at 19:47
Brahadeesh
4,11331550
edited Aug 26 at 19:47
Brahadeesh
4,11331550
edited Aug 26 at 19:47
Brahadeesh
4,11331550
4,11331550
asked Aug 26 at 19:38
Debprasad Kundu
2
asked Aug 26 at 19:38
Debprasad Kundu
2
asked Aug 26 at 19:38
Debprasad Kundu
2
2
closed as off-topic by Shaun, Morgan Rodgers, TheGeekGreek, Arnaud D., Nex Aug 27 at 19:17
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." – Shaun, Morgan Rodgers, TheGeekGreek, Nex
closed as off-topic by Shaun, Morgan Rodgers, TheGeekGreek, Arnaud D., Nex Aug 27 at 19:17
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." – Shaun, Morgan Rodgers, TheGeekGreek, Nex
What about $beginbmatrix1 & 1\ 0 & 1 endbmatrix$?
– Calculon
Aug 26 at 19:41
1
You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Aug 26 at 19:42
Take a look at en.wikipedia.org/wiki/Diagonalizable_matrix It is a good idea to search for answers to questions like this before asking. This way you will understand why all non-singular matrices are not diagonalizable instead of being presented with some useless counter-example.
– John Douma
Aug 26 at 19:48
add a comment |Â
What about $beginbmatrix1 & 1\ 0 & 1 endbmatrix$?
– Calculon
Aug 26 at 19:41
1
You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Aug 26 at 19:42
Take a look at en.wikipedia.org/wiki/Diagonalizable_matrix It is a good idea to search for answers to questions like this before asking. This way you will understand why all non-singular matrices are not diagonalizable instead of being presented with some useless counter-example.
– John Douma
Aug 26 at 19:48
What about $beginbmatrix1 & 1\ 0 & 1 endbmatrix$?
– Calculon
Aug 26 at 19:41
What about $beginbmatrix1 & 1\ 0 & 1 endbmatrix$?
– Calculon
Aug 26 at 19:41
1
1
You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Aug 26 at 19:42
You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Aug 26 at 19:42
Take a look at en.wikipedia.org/wiki/Diagonalizable_matrix It is a good idea to search for answers to questions like this before asking. This way you will understand why all non-singular matrices are not diagonalizable instead of being presented with some useless counter-example.
– John Douma
Aug 26 at 19:48
Take a look at en.wikipedia.org/wiki/Diagonalizable_matrix It is a good idea to search for answers to questions like this before asking. This way you will understand why all non-singular matrices are not diagonalizable instead of being presented with some useless counter-example.
– John Douma
Aug 26 at 19:48
add a comment |Â
1 Answer
1
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up vote
1
down vote
No, not every nonsingular matrix is diagonalisable. For instance, this complex matrix isn't:
$$
A = beginpmatrix
1 & 1 \
0 & 1
endpmatrix.
$$
Since you mention the geometric and algebraic multiplicities in your question details, note that the above matrix is already in Jordan canonical form. If it were diagonalisable, then it would be similar to the matrix
$$
beginpmatrix
1 & 0 \
0 & 1
endpmatrix,
$$
by comparing the eigenvalues. But this is also in Jordan canonical form, and the Jordan canonical form of a matrix is unique (upto ordering of the blocks). This contradiction shows that the matrix $A$ isn't diagonalisable.
answered Aug 26 at 19:40
Brahadeesh
4,11331550
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
No, not every nonsingular matrix is diagonalisable. For instance, this complex matrix isn't:
$$
A = beginpmatrix
1 & 1 \
0 & 1
endpmatrix.
$$
Since you mention the geometric and algebraic multiplicities in your question details, note that the above matrix is already in Jordan canonical form. If it were diagonalisable, then it would be similar to the matrix
$$
beginpmatrix
1 & 0 \
0 & 1
endpmatrix,
$$
by comparing the eigenvalues. But this is also in Jordan canonical form, and the Jordan canonical form of a matrix is unique (upto ordering of the blocks). This contradiction shows that the matrix $A$ isn't diagonalisable.
answered Aug 26 at 19:40
Brahadeesh
4,11331550
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
add a comment |Â
up vote
1
down vote
No, not every nonsingular matrix is diagonalisable. For instance, this complex matrix isn't:
$$
A = beginpmatrix
1 & 1 \
0 & 1
endpmatrix.
$$
Since you mention the geometric and algebraic multiplicities in your question details, note that the above matrix is already in Jordan canonical form. If it were diagonalisable, then it would be similar to the matrix
$$
beginpmatrix
1 & 0 \
0 & 1
endpmatrix,
$$
by comparing the eigenvalues. But this is also in Jordan canonical form, and the Jordan canonical form of a matrix is unique (upto ordering of the blocks). This contradiction shows that the matrix $A$ isn't diagonalisable.
answered Aug 26 at 19:40
Brahadeesh
4,11331550
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
add a comment |Â
up vote
1
down vote
up vote
1
down vote
No, not every nonsingular matrix is diagonalisable. For instance, this complex matrix isn't:
$$
A = beginpmatrix
1 & 1 \
0 & 1
endpmatrix.
$$
Since you mention the geometric and algebraic multiplicities in your question details, note that the above matrix is already in Jordan canonical form. If it were diagonalisable, then it would be similar to the matrix
$$
beginpmatrix
1 & 0 \
0 & 1
endpmatrix,
$$
by comparing the eigenvalues. But this is also in Jordan canonical form, and the Jordan canonical form of a matrix is unique (upto ordering of the blocks). This contradiction shows that the matrix $A$ isn't diagonalisable.
answered Aug 26 at 19:40
Brahadeesh
4,11331550
No, not every nonsingular matrix is diagonalisable. For instance, this complex matrix isn't:
$$
A = beginpmatrix
1 & 1 \
0 & 1
endpmatrix.
$$
Since you mention the geometric and algebraic multiplicities in your question details, note that the above matrix is already in Jordan canonical form. If it were diagonalisable, then it would be similar to the matrix
$$
beginpmatrix
1 & 0 \
0 & 1
endpmatrix,
$$
by comparing the eigenvalues. But this is also in Jordan canonical form, and the Jordan canonical form of a matrix is unique (upto ordering of the blocks). This contradiction shows that the matrix $A$ isn't diagonalisable.
answered Aug 26 at 19:40
Brahadeesh
4,11331550
edited Aug 26 at 19:45
answered Aug 26 at 19:40
Brahadeesh
4,11331550
answered Aug 26 at 19:40
Brahadeesh
4,11331550
answered Aug 26 at 19:40
Brahadeesh
4,11331550
4,11331550
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
add a comment |Â
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
Can I give any condition on non singular matrix (not computing P where p^-1AP=diagonal matrix) so that the matrix can be diagonalizable?
– Debprasad Kundu
Aug 27 at 4:30
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
@DebprasadKundu there are several equivalent criteria for a matrix to be diagonalizable, but I don’t know of any that are specific to non-singular matrices. Maybe this one is useful-? “A non-singular complex matrix is diagonalizable if and only if its minimal polynomial $p(x)$ splits as a product of distinct linear factors and $p$ is not divisible by $x$.â€Â
– Brahadeesh
Aug 27 at 4:34
add a comment |Â
What about $beginbmatrix1 & 1\ 0 & 1 endbmatrix$?
– Calculon
Aug 26 at 19:41
1
You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Aug 26 at 19:42
Take a look at en.wikipedia.org/wiki/Diagonalizable_matrix It is a good idea to search for answers to questions like this before asking. This way you will understand why all non-singular matrices are not diagonalizable instead of being presented with some useless counter-example.
– John Douma
Aug 26 at 19:48