The characteristic polynomial of the matrix $A^k$ is given by
\begin{align*}
p(t)&=\det(A^k-tI)\\
&=\det(S^{-1})\det(A^k-tI)\det(S)\\
&=\det(S^{-1}(A^k-tI)S)\\
&=\det(S^{-1}A^kS-tI)\\[6pt]
&=\begin{vmatrix}
\lambda_1^k-t & * & * & * &*\\
0 &\lambda_2^k-t & * & * &*\\
\vdots & \cdots & \ddots & \cdots& \vdots \\
0 & 0 & 0 & \lambda_{n-1}^k-t & *\\
0 & 0 & 0 & 0& \lambda_n^k-t
\end{vmatrix}\\[6pt]
&=\prod_{i=1}^n(\lambda_i^k-t).
\end{align*}

Since the roots of the characteristic polynomial are all the eigenvalues, we see that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of $A^k$.

If Eigenvalues of a Matrix $A$ are Less than $1$, then Determinant of $I-A$ is Positive
Let $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda<1$.
Then show that the determinant \[ |I-A|>0,\]
where $I$ is the $n \times n$ identity matrix.
We give two solutions.
Solution 1.
Let $p(t)$ be […]

Determinant/Trace and Eigenvalues of a Matrix
Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]

If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent
Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.
Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.
Steps.
Use the Jordan canonical form of the matrix $A$.
We want […]

Nilpotent Matrix and Eigenvalues of the Matrix
An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix.
Prove the followings.
(a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.
(b) The matrix $A$ is nilpotent if and only if […]

A Square Root Matrix of a Symmetric Matrix
Answer the following two questions with justification.
(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.
(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]

Eigenvalues of Squared Matrix and Upper Triangular Matrix
Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix.
If
\[P^{-1}AP=\begin{bmatrix}
1 & 2 & 3 \\
0 &4 &5 \\
0 & 0 & 6
\end{bmatrix},\]
then find all the eigenvalues of the matrix $A^2$.
We give two proofs. The first version is a […]

Finite Order Matrix and its Trace
Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that
(a) $|\tr(A)|\leq n$.
(b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.
(c) $\tr(A)=n$ if and only if $A=I_n$.
Proof.
(a) […]

How to Calculate and Simplify a Matrix Polynomial
Let $T=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &1 \\
0 & 0 & 2
\end{bmatrix}$.
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\]
where $I$ is the $3\times 3$ identity matrix.
(The Ohio State University exam)
Hint.
Use the […]

[…] Let $lambda_1$ and $lambda_2$ be eigenvalues of $A$. Then we have begin{align*} 3=tr(A)=lambda_1+lambda_2 text{ and }\ 5=tr(A^2)=lambda_1^2+lambda_2^2. end{align*} Here we used two facts. The first one is that the trace of a matrix is the sum of all eigenvalues of the matrix. The second one is that $lambda^2$ is an eigenvalue of $A^2$ if $lambda$ is an eigenvalue of $A$, and these are all the ei…. […]

## 1 Response

[…] Let $lambda_1$ and $lambda_2$ be eigenvalues of $A$. Then we have begin{align*} 3=tr(A)=lambda_1+lambda_2 text{ and }\ 5=tr(A^2)=lambda_1^2+lambda_2^2. end{align*} Here we used two facts. The first one is that the trace of a matrix is the sum of all eigenvalues of the matrix. The second one is that $lambda^2$ is an eigenvalue of $A^2$ if $lambda$ is an eigenvalue of $A$, and these are all the ei…. […]