Tagged: Cayley-Hamilton theorem

The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$

Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}
2 & 1\\
1& 2
\end{bmatrix}$.

 
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Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$

Problem 471

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

(b) Let
\[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.

(Kyushu University, Linear Algebra Exam Problem)
 
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Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices

Problem 375

Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ elements.
Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

 
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Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial

Problem 348

Let $A$ be an $n\times n$ complex matrix.
Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers.

Let $C$ be the companion matrix of the polynomial $p(x)$ given by
\[C=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & & \ddots & & \vdots \\
0 & 0 & \dots & 1 & -a_{n-1}
\end{bmatrix}=
[\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],\] where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by
\[\mathbf{a}=\begin{bmatrix}
a_0 \\
a_1 \\
\vdots \\
a_{n-1}
\end{bmatrix}.\]

Then prove that the following two statements are equivalent.

  1. There exists a vector $\mathbf{v}\in \C^n$ such that
    \[\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}\] form a basis of $\C^n$.
  2. There exists an invertible matrix $S$ such that $S^{-1}AS=C$.
    (Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

 
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