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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Describing the space of matrices which “jordanize” a given matrix

This is a naive linear algebra question. I apologize for the level but I could not find an answer in the literature. Let $A$ be a $n$ by $n$ matrix (say over $\mathbb C$). Suppose the Jordan form of $...
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41 views

Show that jordan matrix with k blocks in diagonal has exactly k independent eigenvectors?

How can I prove that a Jordan matrix with k blocks in the diagonal has exactly k independent eigenvectors. Can you help me to find a formal proof of this statement?
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Examples of decomposable nilpotent endomorphisms

Every body knows that if $V$ is the real vector space $\mathbb R[X]_{\leq n}$ of polynomials of degree at most $n$, then the linear map $$D:p\in V\mapsto p'\in V$$ given by derivation is nilpotent. It ...
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1answer
38 views

Direct sums of invariant subspaces

Let $A$ be a complex $n\times n$ matrix, with its Jordan carnonical form as $J=diag(J_1,\cdots,J_s)$. Then there exists an invertible matrix $P$ such that $P^{-1}AP=J$. It is easy to verify that $\Bbb ...
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1answer
94 views

Transformation matrix is jordan normal form

I have the following question: Given a finite-dimensional, unitary vector space V and a endomorphism f on V, is it possible to choose an orthonormal basis B of V in such a way, that the transformation ...
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1answer
67 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar [duplicate]

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A \...
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340 views

Jordan canonical form when field is not algebraically closed

Suppose we have a linear operator $T : V \to V$, where $V$ is a vector space $V$ over a field $F$. Now if $F$ is not algebraically closed, we don't necessarily know that $T$ has a Jordan canonical ...
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169 views

Jordan Normal Form of a self-adjoint Linear Transformation

Let $V$ a finite inner product space, $dim V = n \geq 3$. Let $w_1,w_2 \in V$ such that: $<w_1,w_2>=0$, $||w_1||=||w_2||=1$ where $||w||$ is the norm of a vector $w$. The inner product ...
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1answer
72 views

If $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, find $t$ operators such that $T=T_1\oplus\dots\oplus T_t$

Be $T\in \mathscr{L}(V)$ a linear operator with characteristic polynomial $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, $n_i\geq 1$ and $\lambda_i\neq\lambda_j$ if $i\neq j$. Show that $T$ can ...
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532 views

Complex eigenvalues Jordan real matrix

As I posted here and here I'm studying Jordan forms and similar concepts. I've got a problem with complex eigenvalues in jordan real matrices. I know (at least I think so) how to compute the Jordan ...
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146 views

Finding the Jordan Basis and the Canonical form corresponding to the Jordan basis

\begin{pmatrix} 4 & 0 &0 \\ 2 &1 &3 \\ 5& 0 &4 \end{pmatrix} I know that the Characteristic polynomial is : $$(t-4)^2(t-1)$$ I started with eigenvalues $λ=1$ and got in the ...
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80 views

What does power of a factor in a minimal polynomial mean in rational form?

Let $V$ be a finite dimensional vector space and $T$ be some linear operator. Suppose the minimal polynomial has a factor $(x-c)^2$. If $T$ has Jordan form, then we can assert that the biggest jordan ...
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50 views

Lipschitz continuity of invariant subspaces for parametrized matrices

Let $A(t)$ be a one-dimensional parametrized family of linear operators on $\mathbb{R}^m$ that has smooth dependence on $t$. Let $V_0\subset \mathbb{R}^n$ be an $n$-dimensional invariant subspace for ...
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62 views

How do I interpret “K2 mod K1”?

I've been doing a bit of work these past couple of nights on computing cyclic subspace decompositions, finding cyclic bases, and then computing the Jordan canonical form of matrices. My question is: ...
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1answer
156 views

How to turn this matrix to Jordan normal form?

Matrix $A$ is $ \left( \begin{array}{ccc} 3 & 0 & 8 \\ 3 & -1 & 6 \\ -2 & 0 & -5 \end{array} \right)$ and I need to find a matrix P such that $P^{-1} A P = J$ where $J$ is a ...
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252 views

Relationship Jordan Form and Rational Canonical Form

If $A$ is a matrix over a field whose characteristic polynomial splits, then how is the Jordan form related to the rational Canonical form and can we recover one from the other in a computationally ...
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1answer
322 views

Question about exponential of Jordan matrix?

If J is a Jordan matrix. How to prove that the elements on the principle diagonal of $ e^J$ are exponential function of elements on the principle diagonal of $J$? If it is two dimensional, then I can ...
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141 views

Jordan decomposition algorithm

I'm trying to calculate the value of a matrix function. As far as I understood, this is done by first decomposing my matrix $A$ into $PJP^{-1}$. Where $J$ is in Jordan normal form. However, this ...
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1answer
224 views

Eigenspaces and jordan normal form

I have a question here regarding the jordan normal form of two matrices where the eigenspace is one is contained in the other. Let $A,B$ be two $n \times n$ matrices s.t $AB=BA$. I firstly proved ...
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665 views

About Jordan-Chevalley decomposition

I have this problem: Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$. I saw that this ...
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936 views

Decompose $A=D+N$ with $DN=ND$, $N$ nilpotent, $D$ diagonalizable

Can anyone help me out with the following question: For the matrix $A$ give a diagonalizable matrix $D$ and a nilpotent matrix $N$ so that $A=D+N$ and $ND=DN$. $\begin{bmatrix} 1 & 4 \\ -1 & ...
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1answer
295 views

Finding Jordanizing matrix

Let $$A=\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ And I found B, A's Jordan form to be: $$B=\left(\begin{matrix}0&1&0 \\ 0&0&0\\ 0&...
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2answers
64 views

Finding the Jordan Form of a matrix…

I know that this type of question has been asked on here before but I am still having a hard understanding what is going on. The text that I am learning from is "Linear Algebra Done Right by Sheldon ...
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100 views

What's the Jordan form of $J^2$?

$$J^2=\begin{pmatrix}0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0\\ 0&0&0&0&0\end{pmatrix}$$ What's the Jordan form of ...
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1answer
209 views

Can the jordan canonical form be $[0]$?

Let $$M=\begin{bmatrix}3 & 0 & 2 & 4 \\ 1 & 0 & 4 & 3 \\ 3 & 1 & 0 & 0 \\ 0 & 2 & 1& 2 \\ \end{bmatrix}\in(\mathbb{Z}/\mathbb{5Z})$$ I want to prove ...
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2answers
210 views

Jordan canonical form of a matrix $A$ with the property that $A^2 = 0$?

Suppose $A \in \mathbb{C}^{2 \times 2}$ has the property that $A^2 = 0$. What are the possible Jordan canonical forms of such a matrix? A matrix of the form: \begin{pmatrix} a & a \\ -a &...
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5k views

Calculation of the Jordan canonical form

Given the matrix $ F= \begin{bmatrix} 3 & -1 & 0 \\ 1 & 1 & -2 \\ 0 & 0 & 2 \end{bmatrix}$ calculate the Jordan canonical form such that $F = T F_j T^{-1}$. The ...
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1answer
76 views

Find the jordan form the matrix

let characteristic polynomial $P_A(x)=(x+2)^4(x-3)^2$ and minimal polynomial $m_A=(x+2)^3(x-3)$ find the jordan form that possible. we know $q_6=\frac{f_6}{f_5}$ ($f_i$ is gcd{det of i x i ...
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1answer
116 views

$A^3 = A^2$ How can $A$'s minimal polynomial look like?

Let $K$ be a field and $A \in K^{n \times n}$ a matrix with $A^3 = A^2$. How can $A$'s minimal polynomial $\mu_A$ look like? The only possibilities I could think of are $A = 0$. Then the ...
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2answers
1k views

Jordan Canonical Form. Jordan Normal Form.

Just had a couple of quick questions regarding Jordan Normal Form (JNF). Can all matrices be put into JNF? What is the difference between JNF and Jordan Canonical Form. What are JNF and JCF useful ...
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2answers
78 views

Show that a Jordan block is not a square of any matrix.

How do you show that $$J_n(0)= \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \...
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3answers
928 views

Finding a Jordan normal form and basis for matrix

I want to find a Jordan normal form and bases for matrix: $A = \begin{pmatrix} 1&1&-1\\-3&-3&3\\-2&-2&2 \end{pmatrix}$ But it's characteristic polynomial is -$x^3$ which ...
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2answers
47 views

subtle confusion about the definition of Jordan normal form

When I tried to comprehend the definition of Jordan normal form, I noticed that the Field should contains all eigenvalues of the linear operator. I feel confused that if eigenvalue of linear ...
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2answers
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Action of one element of order $n$ and the $n$-th roots of unity

Let $G$ be a finite group and $\rho : G\to GL(V)$ a representation of $G$ on the vector space $V$. If $g\in G$ is one element of order $n$ we have $g^n=1$, so that $$\rho(g^n)=\rho(g)^n=I,$$ where $...
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2answers
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Suppose the characteristic polynomial is $x^4$. Is it possible to get a jordan block of size$ J(2)J(2)$?

Suppose the characteristic polynomial is $x^4$. Is it possible to get a jordan block of size $ J(2)J(2)$? This would mean that I get $2$ vectors in the eigenspaces, but then it seems like an ...
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2answers
219 views

Determine a Jordan Canonical basis for $T^2$

Question: Suppose that $T : V \to V$ is a linear operator on a complex vector space and that $\{v_1,v_2,\dots,v_n\}$ is a basis for $V$ that is a single Jordan chain (in other words, a cycle of ...
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1answer
712 views

How do I know that an inverse of a matrix has the same type of Jordan canonical form

Let $A$ be an invertible matrix in $M_n(\mathbb{C})$. How do I prove that $A^{-1}$ has the same block structure in its Jordan canonical form as $A$ does? For each $x\in \mathbb{C}^n, A(x)=\lambda x$ ...
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1answer
127 views

How can I compute higher powers of a large matrix quickly?

Is there a block matrix technique, assuming that the matrix has a lot of zeroes in it? I want to compute its nilpotence degree. Thanks, (Even with a lot of zeroes in the matrix, there's still a lot ...
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1answer
87 views

$2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My ...
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2answers
123 views

Finding $\bf{P}$ such that $\bf{P^{-1}AP=B}$ for both fixed $\bf{A},\bf{B}$.

How can I find a matrix $\bf{P}\in \mathbb{{R}^{n\times n}}$, such that $\bf{P^{-1}AP=B}$,where $$\bf{A}=\begin{bmatrix} \bf{A_2}& \bf{C_2}& \\ & \bf{A_2}& \bf{C_2}& \\ ...
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1answer
75 views

Why is Jordan Normal Form unique?

I know that the Jordan Normal Form of a matrix is unique (up to reordering the Jordan blocks), but I don't really see why. Say we're looking at a 3x3 case. Now, all we need to do to compute the Jordan ...
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1answer
77 views

Hoffman and Kunze ,Linear algebra Sec 7.4 exercise 4

Construct a linear operator $T$ with minimal polynomial $ x^2(x-1)^2 $ and characteristic polynomial $x^3(x-1)^4$. Describe the primary decomposition of the vector space under $T$ and find the ...
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1answer
34 views

Convergence of powers of matrix given convergence of the powers of its absolute value.

I have a matrix A and a matrix B such that $B_{i, j} = |A_{i, j}|$. I am given that all of the eigenvalues of B have magnitude less than 1, and therefore: $ \displaystyle \lim_{k \to \infty} B^k = 0$ ...
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1answer
57 views

Let $\;A\;$ be a $\;2\times 2-$matrix with only one eigenvalue $\;x=5.\;$ Show that $\;(5I −A)^2 = 0.$

I know that every matrix is conjugate to an upper triangle form matrix and conjugate matrices have the same characteristic polynomial. I then try to get the characteristic polynomial of the upper ...
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1answer
101 views

Matrix that are not upper triangular

I saw in the book "Linear algebra done right" (by S. Axler) that all complex operator has a Jordan form. The proof is based on the fact that all complex operator is upper triangular (i.e. there is a ...
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2answers
468 views

Determine size or number of jordan blocks

I have done most of the work but I struggle to put this matrix into Jordan Normal Form. $$C=\begin{bmatrix}1 & 0 & 0 & 0&0\\1 & -1 & 0 & 0&-1 \\1 & -1 & 0 &...
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1answer
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Wrong Proof: $\mu_f =\mu_g$ iff $h^{-1}\cdot f \cdot h = g$

Problem Let $V$ be a three-dimensional vector space over $\mathbb{C}$ and let $f,g \in End_{\mathbb{C}}(V)$ with $f^3 = g^3 = 0$. In addition, let $\mu_f$ and $\mu_g$ be the minimal polynomial of $f$ ...
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4answers
52 views

Showing a Matrix is Nilpotent of a certain degree using only properties of matrix multiplication/summations

Suppose I had an $n \times n$ matrix $$N = \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots & 0 & 0\\0 & 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & 1 &...
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3answers
174 views

Finding Jordan-normal form of a special $n\times n$ matrix

Let $$A := \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ -a_n & -a_{n - 1} &...
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1answer
468 views

All possible Jordan normal forms with a given minimal polynomial

Suppose $A \in M_{n\times n}(\Bbb R)$ has minimal polynomial $$(\lambda-1)^2(\lambda+1)^2$$ and $n\leq 6$. Find all possible Jordan normal forms. If we denote the $n \times n$ Jordan block $$J_\...