Linear Algebra

The Subspace of Matrices that are Diagonalized by a Fixed Matrix

MIT Linear Algebra Exam problems and solutions

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.

(MIT-Massachusetts Institute of Technology Exam)

Add to solve later

Hint.

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.

Add to solve later

More from my site

The Vector Space Consisting of All Traceless Diagonal Matrices 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 \\ 0 & 0 & \dots & […]
Find All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix 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 & […]
Subspace of Skew-Symmetric Matrices and Its Dimension 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 $V$. (b) Find the […]
Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite 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 […]
Diagonalizable Matrix with Eigenvalue 1, -1 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 […]
Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. 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 […]
Subspaces of Symmetric, Skew-Symmetric Matrices 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 $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of […]
Hyperplane Through Origin is Subspace of 4-Dimensional Vector Space 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 […]