Characteristic polynomial of a matrix which is partitioned into block matrices
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Let $A$ be an $n times n$ matrix which is symetrically partitioned into upper- or lower-triangular-block matrices labelled
$$A = begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
quad text{or} quad
begin{bmatrix}
A_1 & O \
A_2 & A_3
end{bmatrix}.$$
Then the characteristic polynomial of the matrix $A$ is equal to the product of the characteristic polynomial of $A_1$ and $A_3$.
Edit: Can someone help me prove this?
linear-algebra matrices block-matrices
asked 1 hour ago
mathgeek
11
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up vote
0
down vote
favorite
Let $A$ be an $n times n$ matrix which is symetrically partitioned into upper- or lower-triangular-block matrices labelled
$$A = begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
quad text{or} quad
begin{bmatrix}
A_1 & O \
A_2 & A_3
end{bmatrix}.$$
Then the characteristic polynomial of the matrix $A$ is equal to the product of the characteristic polynomial of $A_1$ and $A_3$.
Edit: Can someone help me prove this?
linear-algebra matrices block-matrices
asked 1 hour ago
mathgeek
11
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You have a question?
– user10354138
1 hour ago
What have you tried? And what is your exact problem?
– Ernie060
1 hour ago
1
We can help you prove this, provided you tell us what you have already attempted. Or are you completely lost? You do know what the characteristic polynomial is in terms of determinants,right?
– астон вілла олоф мэллбэрг
1 hour ago
I do know about the characteristic polynomial of matrices being written in terms of determinants. However, I am not that familiar on block matrices so I am having a hard time trying to prove it.
– mathgeek
43 mins ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $A$ be an $n times n$ matrix which is symetrically partitioned into upper- or lower-triangular-block matrices labelled
$$A = begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
quad text{or} quad
begin{bmatrix}
A_1 & O \
A_2 & A_3
end{bmatrix}.$$
Then the characteristic polynomial of the matrix $A$ is equal to the product of the characteristic polynomial of $A_1$ and $A_3$.
Edit: Can someone help me prove this?
linear-algebra matrices block-matrices
asked 1 hour ago
mathgeek
11
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $A$ be an $n times n$ matrix which is symetrically partitioned into upper- or lower-triangular-block matrices labelled
$$A = begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
quad text{or} quad
begin{bmatrix}
A_1 & O \
A_2 & A_3
end{bmatrix}.$$
Then the characteristic polynomial of the matrix $A$ is equal to the product of the characteristic polynomial of $A_1$ and $A_3$.
Edit: Can someone help me prove this?
linear-algebra matrices block-matrices
linear-algebra matrices block-matrices
asked 1 hour ago
mathgeek
11
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
mathgeek
11
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 1 hour ago
asked 1 hour ago
mathgeek
11
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
mathgeek
11
asked 1 hour ago
mathgeek
11
11
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
mathgeek is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You have a question?
– user10354138
1 hour ago
What have you tried? And what is your exact problem?
– Ernie060
1 hour ago
1
We can help you prove this, provided you tell us what you have already attempted. Or are you completely lost? You do know what the characteristic polynomial is in terms of determinants,right?
– астон вілла олоф мэллбэрг
1 hour ago
I do know about the characteristic polynomial of matrices being written in terms of determinants. However, I am not that familiar on block matrices so I am having a hard time trying to prove it.
– mathgeek
43 mins ago
add a comment |
You have a question?
– user10354138
1 hour ago
What have you tried? And what is your exact problem?
– Ernie060
1 hour ago
1
We can help you prove this, provided you tell us what you have already attempted. Or are you completely lost? You do know what the characteristic polynomial is in terms of determinants,right?
– астон вілла олоф мэллбэрг
1 hour ago
I do know about the characteristic polynomial of matrices being written in terms of determinants. However, I am not that familiar on block matrices so I am having a hard time trying to prove it.
– mathgeek
43 mins ago
You have a question?
– user10354138
1 hour ago
You have a question?
– user10354138
1 hour ago
What have you tried? And what is your exact problem?
– Ernie060
1 hour ago
What have you tried? And what is your exact problem?
– Ernie060
1 hour ago
1
1
We can help you prove this, provided you tell us what you have already attempted. Or are you completely lost? You do know what the characteristic polynomial is in terms of determinants,right?
– астон вілла олоф мэллбэрг
1 hour ago
We can help you prove this, provided you tell us what you have already attempted. Or are you completely lost? You do know what the characteristic polynomial is in terms of determinants,right?
– астон вілла олоф мэллбэрг
1 hour ago
I do know about the characteristic polynomial of matrices being written in terms of determinants. However, I am not that familiar on block matrices so I am having a hard time trying to prove it.
– mathgeek
43 mins ago
I do know about the characteristic polynomial of matrices being written in terms of determinants. However, I am not that familiar on block matrices so I am having a hard time trying to prove it.
– mathgeek
43 mins ago
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
Hint:
$$det(A)=detleft( begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
right)=det(A_1)det(A_3)$$
answered 47 mins ago
Aleksas Domarkas
6355
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint:
$$det(A)=detleft( begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
right)=det(A_1)det(A_3)$$
answered 47 mins ago
Aleksas Domarkas
6355
add a comment |
up vote
0
down vote
Hint:
$$det(A)=detleft( begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
right)=det(A_1)det(A_3)$$
answered 47 mins ago
Aleksas Domarkas
6355
add a comment |
up vote
0
down vote
up vote
0
down vote
Hint:
$$det(A)=detleft( begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
right)=det(A_1)det(A_3)$$
answered 47 mins ago
Aleksas Domarkas
6355
Hint:
$$det(A)=detleft( begin{bmatrix}
A_1 & A_2 \
O & A_3
end{bmatrix}
right)=det(A_1)det(A_3)$$
answered 47 mins ago
Aleksas Domarkas
6355
answered 47 mins ago
Aleksas Domarkas
6355
answered 47 mins ago
Aleksas Domarkas
6355
answered 47 mins ago
Aleksas Domarkas
6355
6355
add a comment |
add a comment |
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You have a question?
– user10354138
1 hour ago
What have you tried? And what is your exact problem?
– Ernie060
1 hour ago
1
We can help you prove this, provided you tell us what you have already attempted. Or are you completely lost? You do know what the characteristic polynomial is in terms of determinants,right?
– астон вілла олоф мэллбэрг
1 hour ago
I do know about the characteristic polynomial of matrices being written in terms of determinants. However, I am not that familiar on block matrices so I am having a hard time trying to prove it.
– mathgeek
43 mins ago