What is an elegant way to find the third eigenvalue of $A=left[begin{smallmatrix} 51&-12 & -21\ 60...
$begingroup$
We had this in an exam today: Given the matrix
$$A=begin{bmatrix}
51&-12 & -21\
60 & -40&28\
57&-68&1
end{bmatrix}$$
Here is the precise question as asked in our exam:
Someone tells you (accurately) that $-48$ and $24$ are eigenvalues of the matrix $A$. Without using a computer, calculator or writing anything down find the third eigenvalue of $A$.
I have used the classical method (through the characteristic polynomial of A) but this leaded to a some tremendous computation and finally got the answer.
But the question intentionally asked to do not make any use of such computation.
How can one elegantly find the third eigenvalue of $A$?
linear-algebra matrices eigenvalues-eigenvectors
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
$endgroup$
add a comment |
$begingroup$
We had this in an exam today: Given the matrix
$$A=begin{bmatrix}
51&-12 & -21\
60 & -40&28\
57&-68&1
end{bmatrix}$$
Here is the precise question as asked in our exam:
Someone tells you (accurately) that $-48$ and $24$ are eigenvalues of the matrix $A$. Without using a computer, calculator or writing anything down find the third eigenvalue of $A$.
I have used the classical method (through the characteristic polynomial of A) but this leaded to a some tremendous computation and finally got the answer.
But the question intentionally asked to do not make any use of such computation.
How can one elegantly find the third eigenvalue of $A$?
linear-algebra matrices eigenvalues-eigenvectors
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
$endgroup$
add a comment |
$begingroup$
We had this in an exam today: Given the matrix
$$A=begin{bmatrix}
51&-12 & -21\
60 & -40&28\
57&-68&1
end{bmatrix}$$
Here is the precise question as asked in our exam:
Someone tells you (accurately) that $-48$ and $24$ are eigenvalues of the matrix $A$. Without using a computer, calculator or writing anything down find the third eigenvalue of $A$.
I have used the classical method (through the characteristic polynomial of A) but this leaded to a some tremendous computation and finally got the answer.
But the question intentionally asked to do not make any use of such computation.
How can one elegantly find the third eigenvalue of $A$?
linear-algebra matrices eigenvalues-eigenvectors
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
$endgroup$
We had this in an exam today: Given the matrix
$$A=begin{bmatrix}
51&-12 & -21\
60 & -40&28\
57&-68&1
end{bmatrix}$$
Here is the precise question as asked in our exam:
Someone tells you (accurately) that $-48$ and $24$ are eigenvalues of the matrix $A$. Without using a computer, calculator or writing anything down find the third eigenvalue of $A$.
I have used the classical method (through the characteristic polynomial of A) but this leaded to a some tremendous computation and finally got the answer.
But the question intentionally asked to do not make any use of such computation.
How can one elegantly find the third eigenvalue of $A$?
linear-algebra matrices eigenvalues-eigenvectors
linear-algebra matrices eigenvalues-eigenvectors
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
edited Dec 15 '18 at 12:18
StubbornAtom
5,98811238
5,98811238
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
asked Nov 23 '17 at 18:49
Guy FsoneGuy Fsone
17.2k42973
17.2k42973
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Use the following property:
Trace of a matrix is equal to the sum of the eigenvalues.
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Use the following property:
Trace of a matrix is equal to the sum of the eigenvalues.
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
add a comment |
$begingroup$
Use the following property:
Trace of a matrix is equal to the sum of the eigenvalues.
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
add a comment |
$begingroup$
Use the following property:
Trace of a matrix is equal to the sum of the eigenvalues.
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
Use the following property:
Trace of a matrix is equal to the sum of the eigenvalues.
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
answered Nov 23 '17 at 18:50
Siong Thye GohSiong Thye Goh
101k1466118
101k1466118
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
add a comment |
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Excellent!${}{}{}$
$endgroup$
– Dave
Nov 23 '17 at 20:17
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
Impressive! Is there a short or intuitive proof for this property?
$endgroup$
– Zacky
Dec 15 '18 at 12:23
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
$begingroup$
here are some proofs.
$endgroup$
– Siong Thye Goh
Dec 15 '18 at 12:39
add a comment |
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