If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 424

Let $A$ and $B$ be $n\times n$ matrices.
Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.
Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.

 
LoadingAdd to solve later
Sponsored Links

Proof.

Since $A$ and $B$ have $n$ linearly independent eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$, they are diagonalizable.
Specifically, if we put $S=[\mathbf{x}_1, \dots, \mathbf{x}_n]$.

Then $S$ is invertible (as column vectors of $S$ are linearly independent) and we have
\[S^{-1}AS=D \text{ and } S^{-1}BS=D,\] where $D$ is the diagonal matrix whose diagonal entries are eigenvalues:
\[D=\begin{bmatrix}
\lambda_1 & 0 & \cdots & 0 \\
0 & \lambda_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \lambda_n
\end{bmatrix}.\] It follows that we have
\[S^{-1}AS=D=S^{-1}BS,\] and hence $A=B$. This completes the proof.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • How to Diagonalize a Matrix. Step by Step Explanation.How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
  • Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary MatrixDiagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix \[A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]
  • Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix \[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]
  • Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is EvenEigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$. Then prove the following statements. (a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number. (b) The rank of $A$ is even.   Proof. (a) Each […]
  • True or False. Every Diagonalizable Matrix is InvertibleTrue or False. Every Diagonalizable Matrix is Invertible Is every diagonalizable matrix invertible?   Solution. The answer is No. Counterexample We give a counterexample. Consider the $2\times 2$ zero matrix. The zero matrix is a diagonal matrix, and thus it is diagonalizable. However, the zero matrix is not […]
  • Diagonalize the Upper Triangular Matrix and Find the Power of the MatrixDiagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
  • Quiz 13 (Part 1) Diagonalize a MatrixQuiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Linear Algebra
Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra
Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors

Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors \[\begin{bmatrix} 1 \\...

Close