Similar Matrices Have the Same Eigenvalues

Linear algebra problems and solutions

Problem 2

Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.

LoadingAdd to solve later

Sponsored Links


Proof.

We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic multiplicities of a matrix are determined by its characteristic polynomial.

Since $A$ and $B$ are similar, there exists an invertible matrix $S$ such that $S^{-1}AS=B$.
Let $p_A(t)$ and $p_B(t)$ denote the characteristic polynomials of $A$ and $B$, respectively.

We have
\begin{align*}
p_B(t) &= \det(B-tI) =\det(S^{-1}AS-tI)  \\
&=\det(S^{-1}(A-tI)S) = \det(S^{-1}) \det(A-tI) \det(S) \\
&\stackrel{(*)}{=}\det(A-tI)=p_A(t).
\end{align*}

Here the fifth equality (*) follows from the fact that $\det(S^{-1})=\det(S)^{-1}$.
(Also note that even though matrix multiplication is not commutative in general but determinants are just numbers, thus we can change the order of the product of determinants.)

Thus we showed that $p_A(t)=p_B(t)$ and this completes the proof.

Related Facts

We proved that if $A$ and $B$ are similar, then their characteristic polynomials are the same.

Since the determinants and the traces are the coefficients of the characteristic polynomials.
Thus, if $A$ and $B$ are similar, their determinants and traces are the same.

See the problem “If two matrices are similar, then their determinants are the same” for a more direct proof of this fact about determinants.


LoadingAdd to solve later

Sponsored Links

More from my site

  • A Matrix Similar to a Diagonalizable Matrix is Also DiagonalizableA Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.   Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
  • Maximize the Dimension of the Null Space of $A-aI$Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
  • Determine a Matrix From Its EigenvalueDetermine 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 […]
  • If Two Matrices are Similar, then their Determinants are the SameIf Two Matrices are Similar, then their Determinants are the Same Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.   Proof. Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that \[S^{-1}AS=B\] by definition. Then we […]
  • Determine Whether Given Matrices are SimilarDetermine Whether Given Matrices are Similar (a) Is the matrix $A=\begin{bmatrix} 1 & 2\\ 0& 3 \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix} 3 & 0\\ 1& 2 \end{bmatrix}$?   (b) Is the matrix $A=\begin{bmatrix} 0 & 1\\ 5& 3 \end{bmatrix}$ similar to the matrix […]
  • Determinant of Matrix whose Diagonal Entries are 6 and 2 ElsewhereDeterminant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere Find the determinant of the following matrix \[A=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & 2 \\ 2 & 2 & 2 & 6 & 2 \\ 2 & 2 & 2 & 2 & 6 \end{bmatrix}.\] (Harvard University, Linear Algebra Exam […]
  • Eigenvalues and their Algebraic Multiplicities of a Matrix with a VariableEigenvalues and their Algebraic Multiplicities of a Matrix with a Variable 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. To find eigenvalues we first compute the characteristic polynomial of the […]
  • All the Eigenvectors of a Matrix Are Eigenvectors of Another MatrixAll the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix Let $A$ and $B$ be an $n \times n$ matrices. Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$. Then prove that each eigenvector of $A$ is an eigenvector of $B$. (It could be that each eigenvector is an eigenvector for […]

You may also like...

4 Responses

  1. Owen says:

    What property allows us to move the tI term from the outside to matrix A at the start of line 2?

    • Yu says:

      Dear Owen,

      You can confirm the calculation as follows. I think it is easier to think backward.

      First $S^{-1}(A-tI)S = S^{-1}AS – S^{-1}(tI)S$ by distributing $S{-1}$ and $S$.
      1
      Now look at the second term $S^{-1}(tI)S$. As $t$ is just a scalar (number), we have

      $S^{-1}(tI)S = t S^{-1}IS = tS^{-1}S= t I$ by the propaty of the identity matrix and inverse matrix.

      Combining these we get the desired equality. Let me know if you have further questions.

  1. 04/26/2017

    […] For a proof, see the post “Similar matrices have the same eigenvalues“. […]

  2. 04/26/2017

    […] We recall that if $A$ and $B$ are similar, then their traces are the same. (See Problem “Similar matrices have the same eigenvalues“.) We compute begin{align*} tr(A)=0+3=3 text{ and } tr(B)=1+3=4, end{align*} and thus […]

Leave a Reply

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

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Idempotent Matrix Problems and Solutions in Linear Algebra
Invertible Idempotent Matrix is the Identity Matrix

A square matrix $A$ is called idempotent if $A^2=A$. Show that a square invertible idempotent matrix is the identity matrix.

Close