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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Computing transformation matrix between similar matrices

If I have the matrix $A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}$ and I've calculated its ...
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In the Jordan-Chevalley decomposition $M=D+N$, how obtaining $D$ and $N$ as polynomials in $M$?

The Jordan-Chevalley expresses a linear operator $M$ as $$M = D + N,$$ where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ ...
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$A$ 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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For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
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Minimal polynomial for any power of Jordan block is same as the minimal polynomial of the Jordan block.

Let $J$ be the $n \times n$ Jordan block corresponding to the eigen value $1$. For any natural number $r$ is it true that the minimal polynomial for $J^r$ is $(X-1)^n$ ? Another way to think about it ...
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Find the Jordan forms of a matrix from just the ranks of its eigenspaces

Let $C$ be a $10\times 10$ matrix whose characteristic polynomial is $(t+2)^5(t-3)^5$. Suppose that $rank((C+2I)^2)=6$ and $rank((C-3I))=8$. What are the possible Jordan forms of $C$? This is the ...
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When is possible to use an orthogonal matrix to put in Jordan form a matrix?

I know that if I have a symmetrical matrix defined on $R$, it is always diagonalisable and I can always find beetwen the matrix of its eigenspaces an orthogonal matrix. While if I have a non ...
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Find the Jordan canonical form [closed]

$N$ is a nilpotent $15\times15$ matrix over $\mathbb{R}$ such that $$\dim(\ker N) = 5, \quad \dim (\ker{N^2}) =8, \quad \dim(\ker{N^3})= 11,$$ $$\dim (\ker{N^4}) = 13, \quad \dim(\ker{N^5}) =15$$ ...
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Does this structured matrix yielding a specified characteristic polynomial admit uniquely defined Jordan blocks?

Let $\mathcal E \subset M_5(\mathbb R)$ be a subset of matrices defined by: for every $A \in \mathcal E$, $A$ admits the structure \begin{align*} A = \begin{pmatrix} 0 & * & 0 & 0 & * ...
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Every eigenvector generating a cyclic subspaces, which contains some generalised eigenvectors .

Is every generalised eigenvector belongs to some cyclic subspace generated by an eigenvector ? I think 'yes' and for JNF , this cyclic subspace be a jordan block corresponding to that generating ...
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Distinguishing between two possibilities for the Jordan normal form of a matrix

Let $A=\begin{pmatrix} 2&2&0&0 \\ -2&-2&0&0 \\ 0&0&-1&1\\0&0&-1&1 \end{pmatrix}$. I noticed that $A^2=0_4$, and I deduced that the characteristic ...
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Center of a non-abelian subgroup of $GL(2, \mathbb{C})$

I'm trying to do the following exercise: Let be $G$ a non-abelian subgroup of $GL(2, \mathbb{C})$. Prove that the center of $G$ is contained in the center of $GL(2, \mathbb{C})$ My (very partial) ...
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What is the difference between ones and zeroes on the superdiagonal in a Jordan normal form matrix?

I've observed that in Jordan normal form the eigenvalues are obviously on the diagonal, but on the superdiagonal there can sometimes be only ones, sometimes be only zeroes and sometimes both. What is ...
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i have to find nth power of matrix which is symmetric about main diagonal and all elements of main diagonals are zero. For example sample matrix $\left( \begin{array}{cc} 0 & 1 & 0 & 0\\ 1 ... 1answer 27 views Verifying understanding of uniqueness of the Jordan form? I would just like to make sure I understand the uniqueness part correctly. Suppose I know that the Jordan blocks$J_1, ..., J_n$make up the Jordan form of$A$. Then I can arrange the blocks$J_1, ...,...
So I just felt the need to refresh my concepts a bit, so I started reading about minimal polynomials. The minimal polynomial $p_A$ for a matrix ${\bf A}$, can be defined as the smallest degree monic ...
Find Jordan Decomposition without calculating $\ker(A - \lambda E)$
Without calculating the null space of $(A - \lambda E)$, find the Jordan decomposition of  A = \begin{pmatrix} 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 &...