# 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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### Prove that $V = \ker(\phi^n) \oplus \text{image}(\phi^n)$

Let $V$ be a $n$-dimensional complex vector space and $\phi:V\to V$ a linear mapping. Prove that $$V = \ker(\phi^n) \oplus \text{image}(\phi^n)$$ Here is my attempt: Since $\phi^n$ is also a linear ...
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### Finding Jordan form

Find Jordan form of the following matrix: $$\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ So I got stuck pretty much trying to find the eigenvalues. ...
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### Finding $P$ in $A = P^{-1}JP$ (Jordan Form)

I'm having a lot of trouble understanding the process of finding a basis for the Jordan canonical form (the "algorithm"). My textbook (Friedberg 4E) isn't very clear, and I can't seem to find anything ...
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### Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
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### Finding the Jordan canonical form of this upper triangular $3\times3$ matrix

I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures. \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 3 \end{...
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### 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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### 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 ...
$A=\begin{pmatrix} -3&-1&1\\ -1&-3&1\\ -2&-2&0 \end{pmatrix}.$ Question: $(i)$ Determine the characteristic equation of A, hence find the eigenvalues of A. $(ii)$ Determine ...
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 ...