Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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4
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1answer
555 views

Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?

Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold? The norm I use ...
3
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2answers
957 views

Why does similarity with a diagonal matrix imply that the Jordan normal form must also be diagonal?

If a matrix representation of a linear transformation is similar to a diagonal matrix, why does this imply that the Jordan normal form must also be diagonal?
3
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1answer
801 views

How to find the 'real' jordan canonical form of a matrix

Given that the the Jordan normal form of a matrix is, $J=\begin{bmatrix}2&1&0&0\\0&2&0&0\\0&0&1-i&0\\0&0&0&1+i\end{bmatrix}$ How do you find the 'real'...
2
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0answers
623 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 ...
2
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1answer
86 views

Prove that $A$ is similar to $A^n$ based on A's Jordan form

Let $A = \begin{bmatrix}1&-3&0&3\\-2&-6&0&13\\0&-3&1&3\\-1&-4&0&8\end{bmatrix}$, Prove that $A$ is similar to $A^n$ for each $n>0$. I found that ...
2
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1answer
179 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 ...
2
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0answers
34 views

Bounding 2-norm of powers of a matrix

Suppose that $A$ is a $n \times n$ matrix with $\rho(A) \leq 1$ and $\|A\|_2 \leq R$, where $R>1$. How can I show an upper bound on $\|A^k\|_2$ that is polynomial in $k$? A trivial upper bound is ...
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1answer
53 views

Find the Jordan form

I am trying to find a matrix $P$ such that $P^{-1}AP$ is the Jordan canonical form. But I am getting confused. I found the characteristic polynomial to be $(x-1)(x-2)^5$ and minimum polynomial to be $(...
1
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2answers
342 views

Jordan form and base for a $n \times n$ Matrix

Given this matrix in size $n \times n$ $\begin{pmatrix} 1 &1 &1&\cdots & 1 & 1 & \\ 0 &1 & & & \\ . & &.& & & \\ . &...
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2answers
298 views

Find the jordan basis for $A$

$$A =\left(\begin{array}{cc}5 & -4 \\9 & -7\end{array}\right)$$ I found that the eigenvalues are $-1$ (algebriac multiplicity 2) Therefore, the jordan form looks like this: $$J =\left(\...
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1answer
464 views

Finding a Jordan Basis after finding the Jordan Canonical Form

The question asked to find the Jordan Canonical Form and Jordan Basis of $\begin{bmatrix}1 & 1 & 0 & -1\\0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\...
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1answer
123 views

Degree of (x-λ) in minimal polynomial

λ is a root of p(x), the minimal polynomial of T (linear operator on complex V). Then λ is an eigenvalue of T. How to prove that the degree of (x-λ) in p(x) equals the size of the largest λ-Jordan ...
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1answer
202 views

Matrix reduction trigonalisaton

Let $ \mathbf{A}=\begin{bmatrix} 2 & -1 & -1 \\ 2 & 1 & -2\\ 3 & -1 & -2 \end{bmatrix} $ Trigonalise a matrix in process of ...
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1answer
1k views

Find the Jordan normal form J for A and a Jordan basis for A.

$A=\begin{pmatrix} -3&-1&1\\ -1&-3&1\\ -2&-2&0 \end{pmatrix}.$ Question: $(i)$ Determine the characteristic equation of A, hence find the eigenvalues of A. $(ii)$ Determine ...
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1answer
173 views

Dimension of centraliser of an $n \times n$ matrix is atleast n.

Let $A$ be an $n \times n$ matrix over a field $\mathbb{F}.$ Then consider the subspace of $M_n(\mathbb{F})$ given by $W=\{ B \in M_n(\mathbb{F}): AB =BA\}.$ Then I want to show that $\text{dim}(W) \...
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2answers
152 views

Jordan normal form (Basis)

Define $A = \begin{pmatrix} -7 & -32 & -32 & -35 \\ 1 & 5 & 4 & 4 \\ 1 & 4 & 5 & 5 \\ 0 & 0 & 0 & 1 \end{pmatrix} \in \mathbb{C^{4x4}}$ I computed ...
5
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2answers
365 views

What exactly determines the block-sizes for Jordan forms?

For instance, with $T \in \mathcal{L}$(Mat($2,2,\mathbb{C}$)) we are given that the minimal polynomial of $T$ is $p(z) = (z - 2i)(z + 7)^2$. I want to find the possible Jordan Forms pertaining to this ...
3
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2answers
257 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 ...
3
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2answers
3k views

How do I write this matrix in Jordan-Normal Form

I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors $v_1=\...
2
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1answer
61 views

Jordan normal form of powers of Jordan normal form

Previous related question: Jordan normal form powers Let $A$ be a $n\times n$ Matrix such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the ...
2
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2answers
561 views

Are the eigenvalues of the matrix AB equal to the eigenvalues of the matrix BA

Are the eigenvalues of the matrix $ AB $ equal to the eigenvalues of the matrix $ BA $ . Where the matrices A And B of sizes $ {3}\mathrm{\times}{5} $ and $ {5}\mathrm{\times}{3} $ Respectively ....
2
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1answer
951 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 ...
2
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1answer
149 views

Find the matrix $S$ such that $A=S^{-1}JS$.

Find the Jordan canonical form $J$ of the matrix:$$A=\begin{pmatrix}0 &0 &0 &0 \\ 0& 0& 0& 0\\ 0 &1& 0& 0\\ 0 &1& 0& 0\end{pmatrix}.$$ Find the matrix $...
2
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2answers
2k views

Jordan block and Jordan chain question?

I don't understand why it's apparently 'clear' that the matrix of $T$ with respect to the basis $v_1, \dots, v_n$ is a Jordan block of degree $n$ if and only if $v_1, \dots ,v_n$ is a Jordan chain for ...
2
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1answer
157 views

Possible Jordan decompositions of stochastic matrices

Which are the possible Jordan normal forms for the stochastic matrices? For some reason I got the idea that they always consist of trivial $1\times 1$ blocks even if eigenvalues of multiplicity $>1$...
2
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1answer
142 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)^{...
2
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1answer
1k 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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2answers
228 views

How to find Jordan basis of a matrix

Assume matrix $$A= \begin{bmatrix} -1&0&0&0&0\\ -1&1&-2&0&1\\ -1&0&-1&0&1\\ 0&1&-1&1&0\\ 0&0&0&0&-1 \end{bmatrix} $$ ...
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1answer
44 views

Finding characteristic and minimal polynomials and the Jordan normal form of $f$, knowing some relations for $f$.

Given a vector space $V$ of dimension $4$ and a base $\{v_1,v_2,v_3,v_4\}$, let $f$ be an endomorphism of $V$ such that $f^3=0$ and moreover $f(v_1)=f(v_2)=v_3$, $f(v_3)=kv_4$, and $f(v_4)\in\left<...
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1answer
180 views

Find Minimal Polynomial and Jordan Normal Form of $T(A)=-A^t$

Let T be a linear transformation such that $T(A)=-A^t$, in the space $M_{n \times n}^{\Bbb R}$, and $-A^t$ is donated as $-A$ Transpose, with the standard inner product. Find $T$'s ...
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1answer
31 views

Jordan normal form powers

Let $A$ be a $n\times n$ such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the eigenvalue and $j$ the order. If $A$ was diagonal($i=1$) ...
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0answers
391 views

Find Jordan form of powers of Jordan matrix

Let $A$ be a Jordan matrix with blocks $J_5(0),J_6(0)$ with $J_m(\lambda)$ having size $m\times m$. I am to find the Jordan form of $A^2$. Since $A$ is in Jordan form and powers of $A$ have the same ...
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0answers
525 views

Primary Rational Canonical form Matrix

As I just posted before on Lost on rational and Jordan forms, I'm practising with jordan forms, rational canonical forms... Well, now I'm stuck in a problem because I don't know where I'm wrong. I ...
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1answer
175 views

Find the Jordan form of this Matrix

M y question is relating to the matrix as A. I have started off this problem by finding the eigenvalues, which turns out to be 3 ( I should note that it has an algebraic multiplicty of 3) From there ...
1
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1answer
81 views

Jordan Cell with rectangular matrices

Suppose $J(\lambda_{1},k_{1})$ and $J(\lambda_{2},k_{2})$ are two Jordan cells and suppose that $C$ is a $k_{1}\times k_{2}$ matrix such that $ J(\lambda_{1},k_{1})C=CJ(\lambda_{2},k_{2}).$ Show that ...
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0answers
177 views

Commutativity and Jordan Decomposition

Apologies in advance if my formatting is bad. I'm working on the following problem and I've run into a bit of a wall. Let $ V $ be a vector space over an algebraically closed field $ \mathcal{F} $....
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2answers
452 views

Normal operator over real inner product space

Let $L: V \rightarrow V$ be a linear operator on a finite dimensional real inner product space $V$ such that $L^{*} = L^{3}$. Show that $L^{2}$ is diagonalizable over $\mathbb{R}$. Attempt: Suppose $...
0
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2answers
153 views

Proving $e^{AT} = e^{\lambda t} \sum_{k=0}^{n-1}\frac{t^{k}}{k!}(A - \lambda I)^{k}$ -final step

Suppose that the Jordan canonical form $J$ of a matrix $A$ is an $n \times n$ Jordan block of the form $$J = \begin{pmatrix} \lambda & 1 & 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 ...
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2answers
74 views

Finding Jordan form of a specific matrix

Let $A\in \text{Mat}_{3\times3}(\mathbb{R})$ such that $A^2-2A+I=0$ and $A\neq I$. Find the Jordan form of $A$.
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0answers
390 views

Decomposing a set of complex matrices into orbits of the operation of conjugation

I need some assistance with the proof for part (b) of the following problem statement: Problem Statement: Decompose the set $\mathbb{C}^{2\times2}$ of $2\times2$ complex matrices into orbits for the ...
0
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1answer
165 views

Finding the Jordan form of a $3\times 3$ matrix

I'm confused: For a matrix with one repeated eigenvalue say $\lambda$, the jordan block for this matrix will look like depending on the nullities of $(A-\lambda I)^n$, doesn't it? I'll give an example:...
0
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1answer
1k views

Finding Jordan Basis of a matrix (3X3)

Having trouble finding the jordan base for this matrix \begin{pmatrix} 1 & 1 &0 \\ 0 &1 &1 \\ 0& 0 &2 \end{pmatrix} I know that the Characteristic polynomial is : (t-1)^2(t-2) ...
0
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1answer
1k views

Jordan normal form for complex matrices

Suppose we are given the  characteristic polynomial and minimal polynomial of a matrix, say, $(x-a)^4(x-b)^2$ and $(x-a)^2(x-b)$. Then, I can tell what the largest Jordan blocks are, and hence work ...

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