Chapter Problems

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ and

Show that $p(A) = 0_{n}$, i.e. the matrix $A$ satisfies its characteristic equation (Cayley-Hamilton theorem).

Let $A$ be an arbitrary $2 \times 2$ matrix. Show that

and therefore $(\operatorname {tr}(A))^{2} = \operatorname {tr}(A^{2}) + 2 \operatorname {det}\left(A\right)$.

Consider the $2 \times 2$ nonnormal matrix

Calculate $f(A) = A^{3}$ using the Cayley-Hamilton theorem.

Apply the Cayley-Hamilton theorem to the $3 \times 3$ matrix $A = (a_{jk})$ and express the result using the trace and determinant of $A$.

Let $A$ be an $n \times n$ matrix. Assume that the inverse matrix of $A$ exists. The inverse matrix can be calculated as follows (Csanky’s algorithm). Let

The roots are, by definition, the eigenvalues $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$ of $A$. We write

where $c_{n} = (-1)^{n} \operatorname {det}\left(A\right)$. Since $A$ is nonsingular we have $c_{n} \neq 0$ and vice versa. The Cayley-Hamilton theorem states that

Multiplying this equation with $A^{-1}$ we obtain

If we have the coefficients $c_{j}$ we can calculate the inverse matrix $A$. Let

Then the $s_{j}$ and $c_{j}$ satisfy the following $n \times n$ lower triangular system of linear equations

Since

we find $s_{k}$ for $k = 1, \ldots , n$. Thus we can solve the linear equation for $c_{j}$. Finally, using the inverse-matrix formula above we obtain the inverse matrix of $A$. Apply Csanky's algorithm to the $3 \times 3$ matrix

Let $A$ be an $n \times n$ matrix. Let

be the characteristic polynomial of $A$. Apply the Cayley-Hamilton theorem $c(A) = 0_{n}$ to calculate $\exp (A)$.

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. Let $f$ be an entire function, i.e. an analytic function on the whole complex plane, for example $\exp (z)$, $\sin (z)$, $\cos (z)$. Using the Cayley-Hamilton theorem we can write

where the complex numbers $a_{0}, a_{1}, \ldots , a_{n-1}$ are determined as follows: Let

which is the right-hand side of the previous equation with $A^{j}$ replaced by $\lambda^{j}$, where $j = 0, 1, \ldots , n-1$ of each distinct eigenvalue $\lambda_{j}$ of the matrix $A$, we consider the equation

If $\lambda_{j}$ is an eigenvalue of multiplicity $k$, for $k > 1$, then we consider also the following equations

Apply this technique to find $\exp (A)$ with

Let $z_{1}, z_{2}, z_{3} \in \mathbb {C}$. Assume that at least two of the three complex numbers are nonzero. Consider the $2 \times 2$ matrix

Calculate $\exp (A)$ applying the Cayley-Hamilton theorem. The characteristic equation for $A$ is given by $\lambda^{2} - z_{1}\lambda - z_{2}z_{3} = 0$ with the eigenvalues

Cayley-Hamilton theorem then states that $A^{2} - z_{1}A - z_{1}z_{3}I_{2} = 0_{2}$.

Let $\alpha \in \mathbb {R}$ and $\alpha \neq 0$. Consider the $2 \times 2$ matrix

Calculate $\sin (A(\alpha ))$ applying the Cayley-Hamilton theorem.

Calculate

Let $A$, $B$ be $n \times n$ matrices such that $ABAB = 0_{n}$. Can we conclude that $BABA = 0_{n}$?