Prove that the matrix
\[A=\begin{bmatrix}
1 & 1.00001 & 1 \\
1.00001 &1 &1.00001 \\
1 & 1.00001 & 1
\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.

(University of California, Berkeley Qualifying Exam Problem)

Let us put $a=1.00001$. We compute the characteristic polynomial $\det(A-tI)$ of the given matrix $A$ as follows.
We have
\begin{align*}
\det(A-tI)&=\begin{vmatrix}
1-t & a & 1 \\
a &1-t &a \\
1 & a & 1-t
\end{vmatrix}\\
&=(1-t)\begin{vmatrix}
1-t & a\\
a& 1-t
\end{vmatrix}-a\begin{vmatrix}
a & a\\
1& 1-t
\end{vmatrix}+\begin{vmatrix}
a & 1-t\\
1& a
\end{vmatrix}
\end{align*}
by the cofactor expansion corresponding to the first row.

Simplifying this, we obtain
\[\det(A-tI)=-t(t^2-3t+2-2a^2).\]
The eigenvalues of $A$ are roots of this characteristic polynomial.
Hence $0$ is an eigenvalue of $A$. Let $\lambda_1$ and $\lambda_2$ be other two eigenvalues of $A$.

Then we have
\[\det(A-tI)=-t(t-\lambda_1)(t-\lambda_2)=-t(t^2-(\lambda_1+\lambda_2)+\lambda_1 \lambda_2).\]
Therefore we have
\[\lambda_1 \lambda_2=2-2a^2.\]

Since $2-2a^2=2(1-a^2)<0$ as $a=1.0001>1$, the product $\lambda_1 \lambda_2$ is negative, and we conclude that one of them is positive and the other is negative.

(Note that if the constant $c$ term of a quadratic polynomial $x^2+bx+c$ is negative, then roots of the polynomial are real and one is negative and the other is positive.)
In summary, the eigenvalues of the matrix $A$ are $0$ and one positive eigenvalue and one negative eigenvalues.

A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
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Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.
Consider the matrix
\[A=\begin{bmatrix}
3/2 & 2\\
-1& -3/2
\end{bmatrix} \in M_{2\times 2}(\R).\]
(a) Find the eigenvalues and corresponding eigenvectors of $A$.
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1 \\
0
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Simple Commutative Relation on Matrices
Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
\[A(A+B)^{-1}B=B(A+B)^{-1}A.\]
(University of California, Berkeley qualifying exam)
Proof.
Let $P=A+B$. Then $B=P-A$.
Using these, we express the given […]

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(b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.
Is it true that the rows of $A$ must also form an orthonormal set?
(University of […]

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Find a square root of the matrix
\[A=\begin{bmatrix}
1 & 3 & -3 \\
0 &4 &5 \\
0 & 0 & 9
\end{bmatrix}.\]
How many square roots does this matrix have?
(University of California, Berkeley Qualifying Exam)
Proof.
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Inequality Regarding Ranks of Matrices
Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
\[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\]
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(University of California, Berkeley, Qualifying Exam)
Hint.
Regard the matrix as a linear transformation $A: […]

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(UCB-University of California, Berkeley, […]

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Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\]
where $a$ is a real number.
Proof.
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