Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
Use the following fact: a scalar $\lambda$ is an eigenvalue of a matrix $A$ if and only if
\[\det(A-\lambda I)=0.\]
Proof.
Since $\lambda$ is an eigenvalue of $A^2$, the determinant of the matrix $A^2-\lambda I$ is zero, where $I$ is the $n \times n$ identity matrix:
\[\det(A^2-\lambda I)=0.\]
Now we have the following factorization.
\begin{align*}
A^2-\lambda I=(A-\sqrt{\lambda} I)(A+\sqrt{\lambda} I).
\end{align*}
Taking the determinant of both sides, we obtain
\begin{align*}
0&=\det(A^2-\lambda I)=\det \left((A-\sqrt{\lambda} I)(A+\sqrt{\lambda} I)\right)\\
&=\det(A-\sqrt{\lambda} I)\det(A+\sqrt{\lambda} I)
\end{align*}
by the multiplicative property of the determinant.
Therefore we have either
\[\det(A-\sqrt{\lambda} I)=0 \text{ or } \det(A+\sqrt{\lambda} I)=0\]
Thus we conclude that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
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 […]
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Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.
Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity […]
How to Find Eigenvalues of a Specific Matrix.
Find all eigenvalues of the following $n \times n$ matrix.
\[
A=\begin{bmatrix}
0 & 0 & \cdots & 0 &1 \\
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 &0\\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & […]
Rotation Matrix in Space and its Determinant and Eigenvalues
For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
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Determine a Matrix From Its Eigenvalue
Let
\[A=\begin{bmatrix}
a & -1\\
1& 4
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is some real number.
Suppose that the matrix $A$ has an eigenvalue $3$.
(a) Determine the value of $a$.
(b) Does the matrix $A$ have eigenvalues other than […]
Similar Matrices Have the Same Eigenvalues
Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.
Proof.
We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]