All Questions
Tagged with characteristic-polynomial matrix-equations
12 questions
2
votes
1
answer
70
views
Does the equation of eigenvalues tell the equation of the matrix?
If $\lambda$'s (eigenvalues) satisfy an polynomial equation, can I derive the same equation for the matrix?
To be specific, if I have $\lambda^2 = 1$, can I get $A^2 = I$?
I know the converse is ...
- 51
0
votes
0
answers
68
views
An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$ [duplicate]
$A$ is an $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$. Then, which of the following is true?
$A^n=A^{n-2}$
rank of $A$ is $2$
rank of $A$ is atleast $2$
there are ...
3
votes
3
answers
84
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How to find ${\rm rank}(2I_n-A)$ where A is a square matrix of size $n$ and $A^3 - 6A^2 + 12A = 0_n$?
I think the rank has to be $n$ since anything else would be impossible to prove with so little information about the matrix.
$$\det(2I_n-A) = -P_A(2) = -\det(A - 2I_n) \ .$$
So, if I can show the ...
- 43
11
votes
1
answer
351
views
If square matrices $A^2 + B^2 = 2AB$, then prove that $p_A(x) = p_B(x)$
Original problem statement:
Let $A, B \in M_n(\mathbb{C})$ such that $A^2 + B^2 = 2AB$. Prove that
for any $x \in \mathbb{C}$: $$det(A - xI_n) = det(B-xI_n)$$
Now the first observation, the equality ...
- 851
2
votes
1
answer
51
views
Proof of a polynomial matrix equation
Consider a $2 \times 2$ matrix
$$ A= \left[\begin{array}{c} 2 & 7\\ 1 & 8\end{array}\right] $$
For this matrix, or for any $2 \times 2$ matrix $A$, why does $A^2 - \mbox{tr}(A) \cdot A + \det(...
1
vote
0
answers
58
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Characteristic polynomial equation for a matrix of a special form
While studying linear algebra, I became interested in the characteristic polynomial.
I identified several types of matrices that interest me and made a system of equations for them.
I am trying to ...
- 95
4
votes
3
answers
1k
views
Can an $n \times n$ matrix satisfy an $n$ degree polynomial equation other than its characteristic polynomial equation?
Can an $n \times n$ matrix satisfy an $n$ degree polynomial equation other than its characteristic polynomial equation?
I was curious if the characteristic polynomial equation is the only $n$ degree ...
- 81
2
votes
2
answers
122
views
Compute a matrix $P$ such that $P$ satisfies $3I+P+P^2=\left(\begin{smallmatrix}3&0&0\\3&6&0\\0&0&6\end{smallmatrix}\right)$
Question: Compute a matrix $P$ such that the matrix $P$ satisfies $3I + P + P^2 = \begin{pmatrix} 3 & 0 & 0\\ 3 & 6 & 0 \\ 0 & 0 & 6 \end{pmatrix}$
Attempt: First, I simplify ...
- 65
1
vote
2
answers
733
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How to show that $det(AB-xI)=det(BA-xI)$ ,for any $x\in \mathbb F$.
There is a problem in Hoffmann Kunze:
Show that $AB$ and $BA$ have the same characteristic polynomials,where $A,B$ are both $n\times n$ matrices in $\mathbb F$.
If $A$ or $B$ is invertible it can be ...
- 3,801
2
votes
2
answers
125
views
Find all $n$ for which there exist A, B, matrices of size n with real entries, such that $A^2B-BA^2=A$.
Find all positive integers $n$ for which there exist two square matrices of size $n$, with real entries, shall we denote them by A and B, such that $A^2B-BA^2=A$ and $A\neq O_n$ and $B\neq O_n$.
So ...
3
votes
0
answers
129
views
About the characteristic equation of a square matrix (Cayley-Hamilton theorem)
Take, for example, $A$ of order $2 \times 2$. The characteristic equation comes to be $$A^2 - (\text{tr}(A)) A + \det (A) = 0 $$ But is this the only quadratic equation that this matrix satisfies? ...
- 1,953
5
votes
2
answers
198
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$ A^2 - B^2 = I_{2n+1} \implies det(AB-BA)=0 $ where A,B are complex matrices of odd size
Let $A, B$ be square matrices (with complex entries) of size $2n+1$, where $n$ is a positive integer.
I need help proving the following: $$A^2 - B^2 = I_{2n+1} \implies det(AB-BA)=0 $$
I've tried ...