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Eigenvalue of an $n times n$ real symmetric matrix with rank 2
Clash Royale CLAN TAG#URR8PPP
up vote
2
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Below is a question from the GATE Exam.
Let A be an $n times n$ real valued square symmetric matrix of rank 2 with
$sum_i=1^n sum_j=1^nA_ij^2=50$. Consider the following statements
(I) One Eigenvalue must be in [-5,5]
(II) The eigenvalue with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigenvalues of A is/are necessarily CORRECT?
(A) Both I and II
(B) I only
(C) II only
(D) Neither I nor II
My attempt:
Let A be $beginbmatrix -5&0\0&5\ endbmatrix$ so, it's eigenvalues are -5 and 5. So statement I is true but II is false. So, the answer is B.
I can understand that the questions ask for conditions which are always true for a real values square symmetric matrix of rank 2.
Is there any better way to solve this?
linear-algebra eigenvalues-eigenvectors
edited Jul 16 at 6:09
copper.hat
122k557156
asked Jul 16 at 5:23
user3767495
857
add a comment |Â
up vote
2
down vote
favorite
Below is a question from the GATE Exam.
Let A be an $n times n$ real valued square symmetric matrix of rank 2 with
$sum_i=1^n sum_j=1^nA_ij^2=50$. Consider the following statements
(I) One Eigenvalue must be in [-5,5]
(II) The eigenvalue with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigenvalues of A is/are necessarily CORRECT?
(A) Both I and II
(B) I only
(C) II only
(D) Neither I nor II
My attempt:
Let A be $beginbmatrix -5&0\0&5\ endbmatrix$ so, it's eigenvalues are -5 and 5. So statement I is true but II is false. So, the answer is B.
I can understand that the questions ask for conditions which are always true for a real values square symmetric matrix of rank 2.
Is there any better way to solve this?
linear-algebra eigenvalues-eigenvectors
edited Jul 16 at 6:09
copper.hat
122k557156
asked Jul 16 at 5:23
user3767495
857
3
Your example only shows that II is not necessarily true. You cannot conclude that the answer is B, because you haven't eliminated D as a possible choice.
– user1551
Jul 16 at 5:29
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Below is a question from the GATE Exam.
Let A be an $n times n$ real valued square symmetric matrix of rank 2 with
$sum_i=1^n sum_j=1^nA_ij^2=50$. Consider the following statements
(I) One Eigenvalue must be in [-5,5]
(II) The eigenvalue with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigenvalues of A is/are necessarily CORRECT?
(A) Both I and II
(B) I only
(C) II only
(D) Neither I nor II
My attempt:
Let A be $beginbmatrix -5&0\0&5\ endbmatrix$ so, it's eigenvalues are -5 and 5. So statement I is true but II is false. So, the answer is B.
I can understand that the questions ask for conditions which are always true for a real values square symmetric matrix of rank 2.
Is there any better way to solve this?
linear-algebra eigenvalues-eigenvectors
edited Jul 16 at 6:09
copper.hat
122k557156
asked Jul 16 at 5:23
user3767495
857
Below is a question from the GATE Exam.
Let A be an $n times n$ real valued square symmetric matrix of rank 2 with
$sum_i=1^n sum_j=1^nA_ij^2=50$. Consider the following statements
(I) One Eigenvalue must be in [-5,5]
(II) The eigenvalue with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigenvalues of A is/are necessarily CORRECT?
(A) Both I and II
(B) I only
(C) II only
(D) Neither I nor II
My attempt:
Let A be $beginbmatrix -5&0\0&5\ endbmatrix$ so, it's eigenvalues are -5 and 5. So statement I is true but II is false. So, the answer is B.
I can understand that the questions ask for conditions which are always true for a real values square symmetric matrix of rank 2.
Is there any better way to solve this?
linear-algebra eigenvalues-eigenvectors
edited Jul 16 at 6:09
copper.hat
122k557156
asked Jul 16 at 5:23
user3767495
857
edited Jul 16 at 6:09
copper.hat
122k557156
edited Jul 16 at 6:09
copper.hat
122k557156
edited Jul 16 at 6:09
copper.hat
122k557156
122k557156
asked Jul 16 at 5:23
user3767495
857
asked Jul 16 at 5:23
user3767495
857
asked Jul 16 at 5:23
user3767495
857
857
3
Your example only shows that II is not necessarily true. You cannot conclude that the answer is B, because you haven't eliminated D as a possible choice.
– user1551
Jul 16 at 5:29
add a comment |Â
3
Your example only shows that II is not necessarily true. You cannot conclude that the answer is B, because you haven't eliminated D as a possible choice.
– user1551
Jul 16 at 5:29
3
3
Your example only shows that II is not necessarily true. You cannot conclude that the answer is B, because you haven't eliminated D as a possible choice.
– user1551
Jul 16 at 5:29
Your example only shows that II is not necessarily true. You cannot conclude that the answer is B, because you haven't eliminated D as a possible choice.
– user1551
Jul 16 at 5:29
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
4
down vote
We are being told that $|A|_F^2 =50$ and also $A$ is a real symmetric matrix with rank $2$.
Hence $sum_i=1^n lambda_i^2 =50$ and we know that the eigenvalues are real.
Suppose all eigenvalues are not in $[-5, 5]$, that is if $|lambda_i|>5, forall i in 1,ldots, n$ . Then, $$sum_i=1^n lambda_i^2 ge n min_i lambda_i^2> 2(5)^2=50.$$
which is a contradiction. Hence $(I)$ is true.
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
add a comment |Â
up vote
2
down vote
Take the matrix$A$ of the form $A= operatornamediag ( 5,5,0,... )$. Then
$|A|_F^2 = 50$, but both eigenvalues are in $[-5,5]$. Hence II is not true.
Since $A$ is symmetric, the singular values are the absolute values of the eigenvalues (which are real). Since $A$ has rank 2 there are exactly two non zero
eigenvalues.
Hence $|A|_F^2 = lambda_1^1+lambda_2^2 = 50$. Hence $min(lambda_1^1,lambda_2^2) le 25$ and so $min(|lambda_1|,|lambda_2|) le 5$
and so $I$ is true.
Hence B is the correct answer.
answered Jul 16 at 5:58
copper.hat
122k557156
add a comment |Â
up vote
0
down vote
If $A = beginbmatrix -5 & 0 \ 0 & 5 endbmatrix $ then $|A|_F^2neq 50.$ If $A $ is a diagonal matrix then $ |D| = max_1 leq i leq n |d_i|$ so $ |A |_F^2 = 5$
A = [-5,0;0,5];
my = norm(A);
display(my)
my =
5
Also, the eigenvalues of a triangular matrix are the diagonal. And diagnonal matrices triangular matrices. So the eigenvalues are $-5,5$
answered Jul 16 at 16:01
RHowe
1,010815
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
We are being told that $|A|_F^2 =50$ and also $A$ is a real symmetric matrix with rank $2$.
Hence $sum_i=1^n lambda_i^2 =50$ and we know that the eigenvalues are real.
Suppose all eigenvalues are not in $[-5, 5]$, that is if $|lambda_i|>5, forall i in 1,ldots, n$ . Then, $$sum_i=1^n lambda_i^2 ge n min_i lambda_i^2> 2(5)^2=50.$$
which is a contradiction. Hence $(I)$ is true.
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
add a comment |Â
up vote
4
down vote
We are being told that $|A|_F^2 =50$ and also $A$ is a real symmetric matrix with rank $2$.
Hence $sum_i=1^n lambda_i^2 =50$ and we know that the eigenvalues are real.
Suppose all eigenvalues are not in $[-5, 5]$, that is if $|lambda_i|>5, forall i in 1,ldots, n$ . Then, $$sum_i=1^n lambda_i^2 ge n min_i lambda_i^2> 2(5)^2=50.$$
which is a contradiction. Hence $(I)$ is true.
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
add a comment |Â
up vote
4
down vote
up vote
4
down vote
We are being told that $|A|_F^2 =50$ and also $A$ is a real symmetric matrix with rank $2$.
Hence $sum_i=1^n lambda_i^2 =50$ and we know that the eigenvalues are real.
Suppose all eigenvalues are not in $[-5, 5]$, that is if $|lambda_i|>5, forall i in 1,ldots, n$ . Then, $$sum_i=1^n lambda_i^2 ge n min_i lambda_i^2> 2(5)^2=50.$$
which is a contradiction. Hence $(I)$ is true.
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
We are being told that $|A|_F^2 =50$ and also $A$ is a real symmetric matrix with rank $2$.
Hence $sum_i=1^n lambda_i^2 =50$ and we know that the eigenvalues are real.
Suppose all eigenvalues are not in $[-5, 5]$, that is if $|lambda_i|>5, forall i in 1,ldots, n$ . Then, $$sum_i=1^n lambda_i^2 ge n min_i lambda_i^2> 2(5)^2=50.$$
which is a contradiction. Hence $(I)$ is true.
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
edited Jul 16 at 5:43
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
answered Jul 16 at 5:38
Siong Thye Goh
77.8k134796
77.8k134796
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
add a comment |Â
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
-What is this $||A||_F^2$ ?
– user3767495
Jul 16 at 9:16
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
Frobenius norm, basically sum of squares of each element.
– Siong Thye Goh
Jul 16 at 9:23
add a comment |Â
up vote
2
down vote
Take the matrix$A$ of the form $A= operatornamediag ( 5,5,0,... )$. Then
$|A|_F^2 = 50$, but both eigenvalues are in $[-5,5]$. Hence II is not true.
Since $A$ is symmetric, the singular values are the absolute values of the eigenvalues (which are real). Since $A$ has rank 2 there are exactly two non zero
eigenvalues.
Hence $|A|_F^2 = lambda_1^1+lambda_2^2 = 50$. Hence $min(lambda_1^1,lambda_2^2) le 25$ and so $min(|lambda_1|,|lambda_2|) le 5$
and so $I$ is true.
Hence B is the correct answer.
answered Jul 16 at 5:58
copper.hat
122k557156
add a comment |Â
up vote
2
down vote
Take the matrix$A$ of the form $A= operatornamediag ( 5,5,0,... )$. Then
$|A|_F^2 = 50$, but both eigenvalues are in $[-5,5]$. Hence II is not true.
Since $A$ is symmetric, the singular values are the absolute values of the eigenvalues (which are real). Since $A$ has rank 2 there are exactly two non zero
eigenvalues.
Hence $|A|_F^2 = lambda_1^1+lambda_2^2 = 50$. Hence $min(lambda_1^1,lambda_2^2) le 25$ and so $min(|lambda_1|,|lambda_2|) le 5$
and so $I$ is true.
Hence B is the correct answer.
answered Jul 16 at 5:58
copper.hat
122k557156
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Take the matrix$A$ of the form $A= operatornamediag ( 5,5,0,... )$. Then
$|A|_F^2 = 50$, but both eigenvalues are in $[-5,5]$. Hence II is not true.
Since $A$ is symmetric, the singular values are the absolute values of the eigenvalues (which are real). Since $A$ has rank 2 there are exactly two non zero
eigenvalues.
Hence $|A|_F^2 = lambda_1^1+lambda_2^2 = 50$. Hence $min(lambda_1^1,lambda_2^2) le 25$ and so $min(|lambda_1|,|lambda_2|) le 5$
and so $I$ is true.
Hence B is the correct answer.
answered Jul 16 at 5:58
copper.hat
122k557156
Take the matrix$A$ of the form $A= operatornamediag ( 5,5,0,... )$. Then
$|A|_F^2 = 50$, but both eigenvalues are in $[-5,5]$. Hence II is not true.
Since $A$ is symmetric, the singular values are the absolute values of the eigenvalues (which are real). Since $A$ has rank 2 there are exactly two non zero
eigenvalues.
Hence $|A|_F^2 = lambda_1^1+lambda_2^2 = 50$. Hence $min(lambda_1^1,lambda_2^2) le 25$ and so $min(|lambda_1|,|lambda_2|) le 5$
and so $I$ is true.
Hence B is the correct answer.
answered Jul 16 at 5:58
copper.hat
122k557156
edited Jul 16 at 6:03
answered Jul 16 at 5:58
copper.hat
122k557156
answered Jul 16 at 5:58
copper.hat
122k557156
answered Jul 16 at 5:58
copper.hat
122k557156
122k557156
add a comment |Â
add a comment |Â
up vote
0
down vote
If $A = beginbmatrix -5 & 0 \ 0 & 5 endbmatrix $ then $|A|_F^2neq 50.$ If $A $ is a diagonal matrix then $ |D| = max_1 leq i leq n |d_i|$ so $ |A |_F^2 = 5$
A = [-5,0;0,5];
my = norm(A);
display(my)
my =
5
Also, the eigenvalues of a triangular matrix are the diagonal. And diagnonal matrices triangular matrices. So the eigenvalues are $-5,5$
answered Jul 16 at 16:01
RHowe
1,010815
add a comment |Â
up vote
0
down vote
If $A = beginbmatrix -5 & 0 \ 0 & 5 endbmatrix $ then $|A|_F^2neq 50.$ If $A $ is a diagonal matrix then $ |D| = max_1 leq i leq n |d_i|$ so $ |A |_F^2 = 5$
A = [-5,0;0,5];
my = norm(A);
display(my)
my =
5
Also, the eigenvalues of a triangular matrix are the diagonal. And diagnonal matrices triangular matrices. So the eigenvalues are $-5,5$
answered Jul 16 at 16:01
RHowe
1,010815
add a comment |Â
up vote
0
down vote
up vote
0
down vote
If $A = beginbmatrix -5 & 0 \ 0 & 5 endbmatrix $ then $|A|_F^2neq 50.$ If $A $ is a diagonal matrix then $ |D| = max_1 leq i leq n |d_i|$ so $ |A |_F^2 = 5$
A = [-5,0;0,5];
my = norm(A);
display(my)
my =
5
Also, the eigenvalues of a triangular matrix are the diagonal. And diagnonal matrices triangular matrices. So the eigenvalues are $-5,5$
answered Jul 16 at 16:01
RHowe
1,010815
If $A = beginbmatrix -5 & 0 \ 0 & 5 endbmatrix $ then $|A|_F^2neq 50.$ If $A $ is a diagonal matrix then $ |D| = max_1 leq i leq n |d_i|$ so $ |A |_F^2 = 5$
A = [-5,0;0,5];
my = norm(A);
display(my)
my =
5
Also, the eigenvalues of a triangular matrix are the diagonal. And diagnonal matrices triangular matrices. So the eigenvalues are $-5,5$
answered Jul 16 at 16:01
RHowe
1,010815
edited Jul 16 at 17:00
answered Jul 16 at 16:01
RHowe
1,010815
answered Jul 16 at 16:01
RHowe
1,010815
answered Jul 16 at 16:01
RHowe
1,010815
1,010815
add a comment |Â
add a comment |Â
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3
Your example only shows that II is not necessarily true. You cannot conclude that the answer is B, because you haven't eliminated D as a possible choice.
– user1551
Jul 16 at 5:29