# 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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### 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 ...
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### 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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### 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'...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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:...
Suppose we are given theĀ  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 ...