Questions tagged [jordan-normal-form]

The Jordan normal form of a matrix is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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59 views

Show that a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$ is similar to $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Let $A$ be a $2 \times 2$ non-zero matrix such that $${A}^{2}=0.$$ How do I find an invertible matrix P such that $${P}^{-1}AP=\begin{bmatrix}0&1\\0&0\end{bmatrix} ?$$ Anyone? Please provide ...
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1answer
35 views

If $A^2=0$ then $J^2=0$ - Jordan Form

If $A$ over the field $\Bbb C$ and $A^2=0$, we can say that $A$ is similar to a jordan form. Does this mean we are able to say that: $$J^2=0$$ aswell, and why?
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2answers
87 views

Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
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1answer
795 views

Classify up to similarity all complex $3 \times 3$ matrices A such that A such that ${A^3} = 2{A^2} - A$

Classify up to similarity all complex $3 \times 3$ matrices A such that A such that ${A^3} = 2{A^2} - A$. Here is what I know: All matrices with complex entries have Jordan canonical forms. If ...
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1answer
233 views

Identify nilpotent matrix according to its characteristic polynomial (all eigenvalues are $0$)

I was wondering about something. Say $A_{n \times n}$ is a matrix and it's characteristic polynomial is $P(x)=x^n$ (all eigenvalues are $0$), can you say that $A$ is a nilpotent matrix? I really don'...
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1answer
269 views

How many Jordan normal forms are there when the characteristic polynomial is $(\lambda+4)^5(\lambda-2)^2$?

Let $A\in M_7(\mathbb{C})$ be a matrix in with the characteristic polynomial $p(A)=(\lambda+4)^5(\lambda-2)^2$. I need to find all Jordan normal forms for this. I think that i can use that the ...
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1answer
30 views

Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$

I have this problem that I am attempting, and am struggling with (b). -- Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$ -- (a) ...
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1answer
61 views

A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$ I considered all possible minimal ...
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1answer
43 views

Find $\dim\ker(A-B)$ if the minimal polynomials of $A$ and $B$ are given

Suppose $A,B$ are $3\times 3$ matrices with minimal polynomials $x^2-4$ and $x+2$ resp. What are the possible dimensions of the kernel of $A-B$? The possible JCFs for $A$ are $(2,2,-2), (2,-2,-2)$. ...
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3answers
63 views

Find all possible Jordan forms

Let $A \in M_3(\mathbb{C})$, and: $\frac1{12}A=[A^2-7A+16I_3]^{-1}$ Find all possible Jordan Forms of A (no need to show different block orders) I was thinking of Algebraic manipulations such as: $...
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1answer
30 views

If there are $6$ matrices in $M_{6,6}(\Bbb C)$ such that they all satisfy $A^2=0$, does this imply that at least two of them are similar?

If there are $6$ matrices in the vector space $M_{6,6}(\Bbb C)$ such that they all satisfy $A^2=0$, does this imply that at least two of them are similar? My approach: Observation: $A^2=0$ implies ...
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1answer
553 views

Inverse of the Jordan block matrix

There is the Jordan block matrix $J_\lambda(n):=\begin{pmatrix} \lambda & 1 & & & \\ & \lambda & 1 \\ & & ... & ... \\ & & & \lambda & 1 \\ & ...
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4answers
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If matrices $A,B$ similar, find nonsingular $S$ s.t.$B=S^{-1}AS$

Consider the matrices below $$A=\begin{bmatrix}9&4&5\\-4&0&-3\\-6&-4&-2\end{bmatrix}$$ and $$B=\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}$$ These ...
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145 views

Can doubly stochastic matrices have non trivial Jordan forms?

This is a followup to a previous question where a nice counter example came up to the proposition "stochastic matrices can only have trivial Jordan forms". This question looks at the more strict case ...
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Jordan Canonical Form set of polynomials over complex field

N: $P_4(C) \to P_4(C)$ defined by $N(p) = p''- 3p'$ Find the canonical form and a canonical basis for the mapping of $N$. I am not sure how to compute this when it is over $\mathbb{C}$ instead of ...
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2answers
92 views

Proving $V=\text{KerT}^n \oplus \text{Im}T^n$ for $V$ in $\mathbb C$

Let $V$ be a vector space over $\mathbb C$ with dimension $n$. Let $T:V\to V$ be a linear transformation. I tried to show that $V=\text{KerT}^n \oplus \text{Im}T^n$ . This is my solution, ...
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1answer
183 views

Finding the n-th power of a matrix using Jordan's normal form

There is a matrix $B$ given $$B = \begin{bmatrix} 5&2&-6\\-5&-2&-6\\4&1&-6\end{bmatrix}$$ I am to find $B^n$ using Jordan's normal form. However I have no idea how to deal with ...
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1answer
57 views

Eigenvalues of matrix

Let $\mathbb{C}^n$ be given. Consider a matrix $A$ on it with one simple eigenvalue zero and all other eigenvalues having strictly negative real part. Now, let $v$ be the eigenvector to eigenvalue $...
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2answers
84 views

Find Jordan's normal form

It is first time I try normally to figure out how it works. So there's the matrix: $$A=\begin{pmatrix} 0 & 6 & 4 & 0 \\ 0 & 4 & 0 & 2 \\ 0 & -1 & 2 & -1 \\1 & ...
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1answer
65 views

Jordan normal form which depends on parameter

Find the values of the parameter $a\in\Bbb R$ for which the Jordan normal form of $$M=\begin{pmatrix}1&1&-1\\0&a-2&4\\0&-1&a+2\end{pmatrix}$$ contains a Jordan block of order $...
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3answers
92 views

Formal solution without handwaving about Jordan normal form

Let $A$ be a $7\times 7$ matrix over $\mathbb C$ with minimal polynomial $(t-2)^3$. I need to prove $\dim \ker (A-2)\geq 3$. The handwavy argument I have is that $\deg m$ is the size of the greatest ...
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1answer
110 views

Jordan Canonical Form of complicated Matrix

$$B:= \left[ \begin{matrix} -1 & 9 &0 &0 &0 \\ 0 & -1 & 0 & 0 & 0 \\ 0&3&-1&0&0 \\ 0 & 1 & 1 & 1 &...
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2answers
103 views

Finding an exponential matrix

How do I find the matrix exponential $e^{tA}$ with $$A = \left(\begin{matrix} 2 & 8 \\ 0 & 2\end{matrix}\right)$$ The eigenvalue is 2 with multiplicity 2, but it yields only 1 eigenvector {$...
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1answer
200 views

Finding Jordan form when characteristic is degree 8 and minimal is degree 4.

I'm hoping someone can help me with this question and see if I understand the theory behind it. Suppose we have a linear transformation $T$ represented by a matrix $A$. Let the characteristic ...
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3answers
134 views

Jordan Normal Form of $T(X)=AX$

Let $T:M^{F}_{n \times n} \to M^{F}_{n \times n} $ be a linear transformation defined by $T(X)=AX$, where $F$ is a field. The matrix $A$ and the transformation $T$ have the same minimal ...
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2answers
209 views

Non-zeroth power of a Jordan block for the eigenvalue $1$ is similar to itself

I'm trying to prove: If $J$ is a single Jordan block corresponding to an eigenvalue $\lambda = 1$, then $J^k$ is similar to $J$, where $k$ is a nonzero integer. Moreover, if $\lambda = 1$ is the only ...
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3answers
128 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that $$A^k=\begin{bmatrix}a^k&0&0\\ka^{k-1}&a^k&...
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1answer
92 views

Find Jordan form for $A = \begin{bmatrix}0&-1\\1&0\end{bmatrix}$ step by step

I'm trying to apply an algorithm I found online to compute the Jordan form of the following matrix $A = \begin{bmatrix}0&-1\\1&0\end{bmatrix}$ Find characteristic polynomial and eigenvalues ...
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2answers
345 views

A Basis for a Jordan Normal Form

In my assignment I have to find a Jordan normal form for this matrix: Thank you for your help, and I'm sorry the question is pronunced with Latex.
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1answer
942 views

Nilpotent Matrices properties

Let $N \in M_n(\mathbb{F})$ be nilpotent. Prove that for any $1 \leq k \in \mathbb{N}$ a matrix $B \in M_n(\mathbb{F})$ exists such that $B^k=I+N$ I have no idea how to het started here. We've just ...
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2answers
1k views

Raising a square matrix to a power n

I have matrix $A$ of $2 \times 2$ and want to raise it to a power of $n$. I was thinking about getting $J$, the Jordan form of $A$ and then separating $J$ to two matrices: $S$ (diagonal matrix) ...
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1answer
63 views

Find a direct way to calculate recursive elements (simple problem with matrices)

I nearly solved this question, I just need a hand with the last part since it is a bit confusing. We are given the recursive sequences $\{a_n\}$ and $\{b_n\}$ like this: $a_1=1$, $b_1=2$ $a_n=a_{n-...
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1answer
130 views

Linear Algebra- Jordan Normal Form

Ok before anyone says Ive posted 3 questions in one post, I would like to state this is given as a single whole question and I'm sure parts before help the parts after. This is a question I'm not able ...
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1answer
59 views

Given the Jordan form $J$, find matrix $P$

In a question set in my linear algebra course, I'm asked the following: Find $P$ such that $P^{-1}AP=J$, where $$A = \begin{pmatrix}6&5&-2&-3\\ -3&-1&3&3\\ 2&1&-2&...
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2answers
50 views

Jordan normal as transformation with respect to the basis of eigenvectors

I have the following matrix $$A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ ...
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1answer
56 views

Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $.

Let $ A, X \in M_{nxn}(K) $. Let $ p_A(t) $ be a characteristic polynomial of matrix A. Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $....
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1answer
39 views

$A$ and $B$ be $n \times n$ matrices over the field $\mathbb F$ which have the same characteristic polynomial

Lemma: Let $N_1$ and $N_2$ be $3 \times 3$ nilpotent matrices over field $\mathbb F$. Then, $N_1$ and $N_2$ are similar if and only if they have the same minimal polynomial. Use the result above ...
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1answer
116 views

Jordan canonical form of matrix that $A^2=A^t$

Good morning can someone help me in this exercise. Let $A\in M(n,\mathbb{R})$ for wich exists an integer $k\ge 1$ such that $A^t=A^k$ where $A^t$ is the trasposed matrix, let also $J(A)$ the jordan ...
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2answers
160 views

Finding Jordan canonical form of a matrix given the characteristic polynomial

I am trying to find the Jordan canonical form of a matrix $A$ given its characteristic polynomial. Suppose $A$ is a complex $5\times 5$ matrix with minimal polynomial $X^5-X^3$. The end goal of the ...
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1answer
55 views

Determine the Jordan normal form of a complex matrix

I want to determine the Jordan normal form of a complex matrix $A$ with characteristic polynomial $\chi_A(x)=(x+1)^4(x+2)^2$, minimal polynomial $m_A(x)=(x+1)^2(x+2)^2$ and that has the property that ...
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1answer
115 views

How many different structures of Jordan forms in $M(n,\mathbb{C})$?

The possible Jordan forms for $n \times n$ matrices in $\mathbb{C}$ can be found if we know that the characteristic polynomial is: $$ p(x)=(x-\lambda_1)^{k_1}(x-\lambda_2)^{k_2}\cdots (x-\lambda_m)^{...
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2answers
457 views

How can I find the dimension of an eigenspace?

I have the following square matrix $$ A = \begin{bmatrix} 2 & 0 & 0 \\ 6 & -1 & 0 \\ 1 & 3 &-1 \end{bmatrix} $$ I found the eigenvalues: $2$ with algebraic and geometric ...
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1answer
51 views

Generalized Eigenvectors when algebraic multiplicity greater than 1

Find the Generalized Eigenvectors of $$ \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 1 & -1 & 0 & 0 & -1\\ 1 & -1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & ...
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1answer
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Jordan canonical form and reordering

Consider the Jordan canonical form below \begin{equation*} J = \begin{pmatrix} J_2(\lambda_1)&0& 0& 0\\ 0&J_1(\lambda_2) &0& 0\\ 0& 0& J_3(\lambda_1)& 0\\ 0 &0 ...
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1answer
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If $\mathrm{Tr} (X)=0$ then there exists $A, B\in \mathcal {M}_n (\mathbb {C}) $ s.t $ X=A\cdot B-B\cdot A $.

Let $X\in \mathcal {M}_n (\mathbb {C}) $ with $\mathrm{Tr}(X)=0$. Then there exists $A, B\in \mathcal {M}_n (\mathbb {C}) $ s.t $X=A\cdot B-B\cdot A $. It's there a solution that uses Normal Jordan ...
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1answer
193 views

Jordan form, number of blocks.

Suppose I have an eigenvalue $\lambda$, now I want to determine the number of Jordan blocks corresponding to that eigenvalue, as well as size of each block. I know that: 1) number of blocks is equal ...
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2answers
76 views

Jordan normal form 33 [closed]

what is the Jordan normal form for matrix $$\begin{bmatrix} 3&1& 0\\0& 3& 0\\0& 0& 2\\ \end{bmatrix}$$ can't figure out because some eigenvalues makes some rows 0 and how ...
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1answer
182 views

Given nullities, how would I determine the Jordan form of a matrix $A$?

Given a $12 \times 12$ matrix with only one eigenvalue $\lambda = 7$, if we're given that $nullity (A-7I) = 4, nullity (A-7I)^2 = 7, nullity (A-7I)^3 = 10, nullity (A-7I)^4 = 12$ how would I find a ...
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1answer
290 views

Finding a Chain Basis and Jordan Canonical form for a 3x3 upper triangular matrix

I have my matrix A, which is $$A=\begin{bmatrix} 3 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \\ \end{bmatrix}$$ and I have been instructed to find a chain basis for it and then put ...
2
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1answer
248 views

Finding the Jordan Canonical Form of a Classical Adjoint of a Jordan Block

Let $A$ be a size $n$ Jordan matrix with $0$ on its diagonal, that is $$A = J_n(0) = [a_{ij}] = \begin{cases} 1, &j=i+1\\ 0, &\text{elsewhere} \end{cases} $$ What is the Jordan Canonical ...