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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Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space $V/\...
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2k views

Finding Jordan basis of a matrix (4x4)

I'm facing a problem finding a Jordan basis for this ($4 \times 4$) matrix: $$\left(\begin{matrix}3&-1&1&7\\9&-3&-7&-1\\0&0&4&-8\\0&0&2&-4\end{matrix}\...
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1answer
95 views

Not quite understanding proof of theorem 2, section 58 of Halmos' Finite-Dimensional Vector Spaces

The proof of theorem 2 of section 58 (Jordan Form) of Halmos' Finite-Dimensional Vector Spaces says the following (this is near the end of the first paragraph in page 114): Since the only proper ...
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1answer
170 views

Left shift operator - Jordan form

Let $S$ be the left shift operator over the space of infinite sequences of complex numbers. Describe the kernels of the operators $S-I$, $(S-I)^2$, $(S-I)^3$. I started by doing the matrices of $S$ ...
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1answer
107 views

Decomposing $\mathbb{R}^{8}$ Using an $8\times 8$ matrix

Let $A$ an $8\times 8$ real matrix such that $A^{21}=I$. Prove that $\mathbb{R}^{8}$ can be written as the direct sum of $4$ two dimensional vector subspaces invariant under $A$, that is $\mathbb{R}^{...
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How to put a matrix in Jordan canonical form, when it has a multiple eigenvalue?

Put the matrix $$\begin{bmatrix} 3 & -4\\ 1 & -1\end{bmatrix}$$ in Jordan Canonical Form. Moreover, find the appropriate transition matrix to the basis in which the original matrix assumes ...
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1answer
524 views

How to put this matrix in Jordan Canonical Form

Suppose you have the matrix $A = \begin{bmatrix} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$ Put it into ...
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minimal polynomial of a matrix B given minimal polynomial of $B^2$

If we are given a minimal polynomial for a matrix $B^2$ can we deduce the minimal polynomial for $B$ $?$ Example: if the minimal polynomial for $B^2$ is $m(\lambda) = \lambda^4$ then can we deduce ...
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1answer
89 views

Using Jordan Normal Form to determine when characteristic and minimal polynomials are identical

Say I want to immediately write down a matrix with an identical minimal and characteristic polynomial. Say, $$ (t-1)^{3}(t-2). $$ My first instinct is to write down Jordan Blocks in a block ...
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59 views

Find the Jordan Canonical Form of the given transformation

The transformation here is $T(f(x)) = f(x + 1) + f(x − 1)$ which is a linear endomorphism on V, where $V={f(x) ∈ R[x] : deg f(x) ≤ 2017}$ So I have to find the jcf J of T. Along with a basis of B. $...
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226 views

Minimal polynomial and possible Jordan forms

Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
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143 views

Classification of bilinear forms: operator $A^{-1} A^T$ for bilinear form $A$

I would like to understand a classification of non-degenerate (not necessary symmetric or skew-symmetric) bilinear forms over an algebraically closed field via an operator $\kappa=A^{-1} A^T$ for a ...
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1k views

Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan blocks....
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Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which $\mathcal{M}_{\mathcal{S}...
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813 views

$A^3 = I$. Find the possible Jordan Forms???

If $A^3 = I$, then I want to find the possible Jordan forms of the matrix. Since the minimal polynomial has degree at most three, each block is at most 3, and the eigenvalues are third roots of unity....
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260 views

On the rank inequality $\operatorname{rank}(A)+\operatorname{rank}(A^3)\geq2\operatorname{rank}(A^2)$

So, I have to show that for all square matrices $A$, $$\newcommand{\rank}{\operatorname{rank}}\rank(A)+\rank(A^3)\geq2\rank(A^2).$$ My attempt at it so far: So, if I try to use this theory: Let $A$ ...
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3answers
174 views

Intuition of generalized eigenvector.

I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
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2answers
635 views

From nilpotent matrix to Jordan matrix

Let's say I have a matrix $A\in M_{n\times n}(\mathbb F)$, and it's given that $A$ is a nilpotent matrix. How Can I find a Jordan matrix that is similar to $A$? because $A$ is nilpotent, I can find a ...
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2answers
685 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?
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1answer
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If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?

We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm. Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 \\ ...
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1answer
303 views

Determine all possible Jordan forms of the $2\times 2$ matrices satisfying in $p(x)=x^3-6x^2+11x-6$

Let $p(x)=x^3-6x^2+11x-6$. Describe the Jordan Canonical forms of the $2\times 2-$ matrices $M$ that satisfy $p(M)=0$ if any such matrices exist. $p(x)=(x-1)(x-2)(x-3)$. so the possible eigenvalues ...
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4answers
92 views

How to prove that A and B are similar

Let be $$A=\begin{pmatrix} \frac{-3}{2} & 2 & \frac{-1}{2} \\ \frac{-1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & -2 & \frac{3}{2} \end{pmatrix}, B=\begin{pmatrix} 0 & 1 & ...
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2answers
143 views

Do linear operators that map one space into a different space have a Jordan canonical form?

I know that this answer is most likely "yes", and that, in the setting of matrices, all matrices are similar to its Jordan form, which is unique (up to the ordering of the Jordan blocks.) But what ...
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2answers
818 views

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$. I appreciate your hints, Thanks
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238 views

Jordan form of a matrix through intuition?

Find the normal form of the matrix $A$: $$\begin{bmatrix}2 & 0 & 0 & 0\\1 & 2 & 0 & 0\\0 & 0 & 1 & 1\\0 & 0 & 0 & 1\end{bmatrix}$$ It looks like A's ...
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3answers
68 views

find the Jordan basis of a matrix

I'm trying to find the Jordan basis of the matrix $$A =\begin{bmatrix} 8 & 1 & 2 \\ -3 & 4 & -2\\ -3 & -1 & 3\end{bmatrix}$$ I've got the characteristic equation to be $CA(x) = ...
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2answers
56 views

Jordan form of a matrix confusion

I was just wondering how I would go about finding the Jordan form of a matrix. I have this matrix here... $\left( \begin{array}{ccc} 1 & 0 & -4\\ 0 & 3 & 0 \\ -2 & 0 & -1 \\ ...
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2answers
158 views

Find an invertible matrix $S$ and a matrix $J$ in Jordan form, such that $S^{-1}AS = J$

$A = \begin{pmatrix} 0 & -1 & 0 & 0 \\ 2 & 0 & -1 & 0 \\ 3 & -1 & -2 & -1 \\ -1 & 0 & 1 & 1 \\ \end{pmatrix}$ ...
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1answer
56 views

linear Algebra Jordan normal form

How to find Jordan normal form $(J_k(λ))^n,n ∈ N$ for $λ \neq 0$ Or at least for $(J_k(λ))^n,n ∈ N$
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2answers
421 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 ....
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2answers
56 views

Jordan Normal Form in $\mathbb F_2$ and $\mathbb F_4$

I want to determine the Jordan normal form of the following matrix: $$\begin{bmatrix}0 & 1\\1 & 1\ \end{bmatrix}$$ in $M_2(\mathbb F_2) $ and $M_2(\mathbb F_4) $. I know how to generally ...
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89 views

$A$ is a complex nilpotent $15X15$ matrix of order 5 ($A^5=0$), $\operatorname{rank}(A)=10$, $\operatorname{rank}(A^3)=4$. Find all possible JCFs

I know how to use every detail except for the rank of $A^3$: A is nilpotent so all the eigenvalues are zero. I know that because the minimal polynomial is $\lambda^5$, the largest block is of size 5 $...
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2answers
38 views

Jordan form of the “multiplicative table” matrix

I have to find the Jordan form of the $(10\times10)-$matrix $A$ with the $n$th row formed by $n(1,2,3,4,5,6,7,8,9,10), \ \ 1 \leq n \leq 10$ I have calculated the determinant of $(A-xI)$ using ...
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3answers
822 views

find the Jordan form and $P$ such that $P^{-1}AP = J$.

Consider the matrix $$A = \left(\begin{array}{cccc} -11&0&-9\\32&1&24\\16&0&13 \end{array}\right)$$ I want to find the Jordan form of $A$, with $1$-s at the bottom and the ...
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2answers
229 views

Jordan matrix (de)composition

So any square matrix $A$ can be decomposed into $A = S J S^{-1}$ where $J$ has a normal Jordan form, moreover $A$ and $J$ are similar matrices. My question is quite straightforward. Given arbitrary ...
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2answers
116 views

A non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root

Is there any quick argument to show that every non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root? Just the existence without computing it. Knowing that $A\in \mathbb C^{2 \times2}$ ...
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1answer
83 views

Jordan similar matrix

I have matrix $B = \begin{bmatrix}1 & 1 & -2 & 0\\2 & 1 & 0 & 2 \\ 1 & 0 & 1 & 1 \\ 0 & -1 & 2 & 1\end{bmatrix}$. I found the characteristic polynomial $...
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1answer
95 views

Is this Jordan decomposition possible?

Is this Jordan form possible? $$J=\begin{pmatrix} \lambda & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & \lambda & 0 & 0 & 0 & 0 & 0\\ 0 &...
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3answers
48 views

Two possible cases for JNF of a matrix

I'm trying to find the Jordan Normal Form of the following matrix: $\pmatrix{2 & 0 & 1 & 1 \\0 & 2 & 1 & 1\\0 & 0 & 2 & 1\\0 & 0 & 0 & 2\\}$. Now since ...
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2answers
55 views

How to decide the Jordan normal form of a matrix?

I have the following matrix: \begin{bmatrix}1&1&0\\-1&3&0\\-1&1&2\end{bmatrix} My matrix has characteristic polynomial $(X-2)^3$ and minimal polynomial $(X-2)^2$. How do I ...
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2answers
257 views

Find Jordan canonical form and basis of a linear operator.

Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear operator such that: $T(x,y,z)=(-y-2z,x+3y+z,x+3z)$, I need to find a Jordan canonical form and a basis. This is what i did: In the first place, I ...
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1answer
97 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 $>1$...
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1answer
83 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 ...
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2answers
85 views

Finding the Jordan form for a $4\times4$ matrix

$$A:= \begin{bmatrix}4 & -4 & -11 & 11\\3 & -12 & -42 & 42\\ -2 & 12 & 37 & -34 \\ -1 & 7 & 20 & -17 \end{bmatrix}$$ I'm struggling with this matrix: ...
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1answer
153 views

Question about conjugate Jordan Form matrices.

This was an example done in class last week. I'm struggling with the details of the argument, but have included my questions at the very end. Let $(t-4)^7$ be the characteristic polynomial for a ...
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1answer
306 views

Determine all possible Jordan forms

Let B be a 10 x 10 matrix and let $\lambda$ be a scalar. Suppose it is known that null$(B-\lambda I)=5$, null$(B-\lambda I)^2=8$ null$(B-\lambda I)^3=9$ Based on this information I am required to ...
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2answers
2k views

Size of Jordan block

Imagine that I'm writing the Jordan form of a matrix and I know that the eigenvalue needs to appear 4 times in the diagonal (algebraic multiplicity is 4) and we need 2 Jordan blocks (geometric ...
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1answer
4k views

Exponential of a Jordan block [duplicate]

I am having difficulties with calculating exponential of a Jordan block, I cannot understand the method, can please someone help me, I have an exam on Monday. 'J' is my Jordan matrix and 'P' is my ...
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1answer
277 views

Commutativity in terms of the Jordan Normal Form.

Let us consider requirements for commutativity of matrices in terms of the Jordan Normal Form, Say we have two matrices $\bf A$ and $\bf B$. Then ${\bf A} = {\bf S}^{-1}{\bf JS}$, where $\bf J$ can be ...