The Subspace of Matrices that are Diagonalized by a Fixed Matrix
Problem 33
Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.
Check the following criteria for a subset to be a subspace.
Theorem. (Subspace criteria)
A subset $W$ of a vector space $V$ is a subspace if and only if
The zero vector in $V$ is in $W$.
For any vectors $A, B \in W$, the addition $A+B\in W$.
For any vector $A\in W$ and a scalar $c$, the scalar multiplication $cA\in W$.
Proof.
Let $V$ be the vector space of all $3$ by $3$ matrices.
Define the subset of $V$
\[ W=\{ A\in V \mid S^{-1}AS \text{ is diagonal.} \}. \]
We want to show that $W$ is a subspace of $V$.
To do this, it suffices to show the following subspace criteria;
The zero vector in $V$ is in $W$.
For any vectors $A, B \in W$, the addition $A+B\in W$.
For any vector $A\in W$ and a scalar $c$, the scalar multiplication $cA\in W$.
The zero vector in the vector space $V$ is the $3$ by $3$ zero matrix $O$.
Since $S^{-1}OS=O$, we have $O\in W$.
To show the second criterion, take $A, B \in W$.
Then we have $S^{-1}(A+B)S=S^{-1}AS+S^{-1}BS$ and since $A, B \in W$, the matrices $S^{-1}AS, S^{-1}BS$ are diagonal.
The addition of diagonal matrices is still diagonal, thus $S^{-1}(A+B)S$ is diagonal and $A+B \in W$.
We finally check the third criterion.
Take $A \in W$ and let $c$ be a scalar.
Then we have $S^{-1}(cA)S=cS^{-1}AS$. The matrix $S^{-1}AS$ is diagonal since $A \in W$.
Since a scalar multiplication of a diagonal matrix is still diagonal, we conclude that the matrix $S^{-1}(cA)S$ is diagonal and $cA\in W$.
Therefore the subset $W$ of $V$ satisfies the criteria to be a subspace.
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Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,
\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
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Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
10001 & 3 & 5 & 7 &9 & 11 \\
1 & 10003 & 5 & 7 & 9 & 11 \\
1 & 3 & 10005 & 7 & 9 & 11 \\
1 & 3 & 5 & 10007 & 9 & 11 \\
1 &3 & 5 & 7 & 10009 & 11 \\
1 &3 & 5 & 7 & 9 & […]
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Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)
(a) Prove that the subset $W$ is a subspace of $V$.
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Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
(a) Prove that $A$ is invertible.
(b) Prove that $A^{-1}$ is symmetric.
(c) Prove that $A^{-1}$ is positive-definite.
(MIT, Linear Algebra Exam Problem)
Proof.
(a) Prove that $A$ is […]
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Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.
Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.
(Stanford University Linear Algebra Exam)
See below for a generalized problem.
Hint.
Diagonalize the […]
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Let
\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic […]
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Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
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Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+3y+5z+7w=0.\]
Then prove that the set $S$ is a subspace of $\R^4$.
(Linear Algebra Exam Problem, The Ohio State […]