3
Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form
$$\begin{bmatrix}
0 & -I \\
I & 0 \\
\end{bmatrix}$$
where $I$ is the $k \times k$ identi...
1
Let $A$ be an $n \times n$ matrix with real entries. Let $Τ$ be the linear operator on $R^n$ which is represented by $A$ in the standard ordered basis, and let $U$ be the linear operator on $C^n$ which is represented by $A$ in the standard ordered basis.
Prove that every $0\neq v \in V$ is a cy...
In mathematics, in linear algebra, a cyclic subspace is a certain special subspace of a finite-dimensional vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.
== Definition ==
Let
T
:
V
→
V
...
