Table of Contents

## Do AB and BA have same eigenvalues?

In other words AB and B A have the same eigenvalues. nonzero eigenvalue of AB has the same geometric multiplicity as it has as an eigenvalue of BA. This may not be true for a zero eigenvalue.

## What is the eigenvalue of AB?

A and B could have all zero eigenvalues while 1 is an eigenvalue of AB: A = [ 0 1 0 0]and B = [ 0 0 1 0]; then AB = [ 1 0 0 0]and A + B = [ 0 1 1 0] . For the same reason, the eigenvalues of A + B are generally not λ + β. Here λ + β = 0 while A + B has eigenvalues 1 and −1.

## How do you find the eigenvalues of a matrix A?

Once the eigenvalues of a matrix (A) have been found, we can find the eigenvectors by Gaussian Elimination. to row echelon form, and solve the resulting linear system by back substitution. – We must find vectors x which satisfy (A − λI)x = 0. – First, form the matrix A − 4I: A − 4I = −3 −3 3 3 −9 3 6 −6 0 .

## Is AB a BA polynomial?

Then the characteristic polynomials of AB and BA are the same. BA are similar which certainly implies AB and BA have the same characteristic polynomial. then AB = B = 0, but BA = 0 so that AB and BA are not similar.

## Are AB and BA the same?

The rank sequences of AB and BA eventually become the same constant (the sum of the ranks of their invertible Jordan blocks). AB and BA are similar if and only if they have the same rank sequences. Here are some other useful known facts. If rank ( A B ) = rank ( B A ) = rank ( A ) , then A B ∼ B A .

## Which is the eigenvector of A and B?

Proof. (a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$. (b) Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to $\mathbf{x}$. Proof. (a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.

## Is the eigenvalues of AB the same as those of Ba?

It is true that the eigenvalues (counting multiplicity) of AB are the same as those of BA. This is a corollary of Theorem 1.3.22 in the second edition of “Matrix Analysis” by Horn and Johnson, which is Theorem 1.3.20 in the first edition.

## Can you predict the eigenvalues of a matrix?

If n × n matrix A has eigenvalues 1, − 1 and n × n matrix B also has eigenvalues 1, − 1, can I then say something about eigenvalues of A B and B A? In general, you can almost never predict the eigenvalues of a product based on the eigenvalues of the matrices you are multiplying together.

## Are the eigenvalues of a Hermitian matrix real?

Eigenvalues of a Hermitian Matrix are Real NumbersShow that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem) We give two proofs.