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 for any nonsingular matrix $A$ there exist $X$ such that $X^2=A$

Prove that given any matrix A, where $$\det(A)\neq0$$ $$A\in M_{n,n}(\mathbb C)$$ the following equation $$X^2=A$$ always has a solution. Should I do something with Jordan Normal form? Any help will ...
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Similar Matrices and their Jordan Canonical Forms [duplicate]

Let $A$ and $B$ be two matrices in $M_n$. Is the following ture: $A$ and $B$ are similar $\iff$ $A$ and $B$ have the same jordan canonical form. Could someone explain?
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If N is elementary nilpotent matrix, show that N Transpose is similar to N

If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ ...
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The index of nilpotency of a nilpotent matrix

Let $A$ a matrix in $\mathcal{M}_5(\mathbb C)$ such that $A^5=0$ and $\mathrm{rank}(A^2)=2$, how prove that $A$ is nilpotent with index of nilpotency $4$? Thanks in advance.
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If $(A-2I)^3(A+2I)^2=0$, then what are the possible Jordan canonical forms of $A$?

Here is the exercise: Let $A$ be a $5\times5$ complex matrix such that $(A-2)^3(A+2)^2=0$, where we define $A-\mu:=A-\mu I$ for scalar $\mu$. Assume that $\lambda=2$ is an eigenvalue of $A$ and its ...
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Two different definitions of Jordan canonical form

I am currently reading two linear algebra books. One is Hoffman/Kunze's and the other one is Friedberg/Insel/Spence's. They define Jordan canonical form of linear operator in different ways. In ...
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Lost on rational and Jordan forms

I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
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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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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 ...
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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 ...
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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 ...