- #1

Shelnutt2

- 57

- 0

## Homework Statement

Find all n x n-matrices A such that Q

^{-1}A Q = A for all invertible n x n matrices Q.

## Homework Equations

None

## The Attempt at a Solution

Proof by induction

Assumption: True for A = k(I

_{n}) where k is a real number

Base case: k = 1

When k =1

A = 1 * I

_{n}= I

_{n}

Q

^{-1}* I

_{n}* Q = I

_{n}

(Q

^{-1}* I

_{n}) * Q = I

_{n}

(Q

^{-1}) * Q =I

_{n}

I

_{n}= I

_{n}.

When k = m

A = m * I

_{n}

Q

^{-1}* (m *I

_{n}) * Q = m * I

_{n}

m * (Q

^{-1}* I

_{n}) * Q = m * I

_{n}

m * (Q

^{-1}) * Q = m * I

_{n}

m * I

_{n}= m * I

_{n}.

m * I

_{n}+ I

_{n}= I

_{n}* (m + 1) =

When k = m+1

A = m+1 * I

_{n}

Q

^{-1}* ((m + 1) *I

_{n}) * Q = (m + 1) * I

_{n}

(m + 1) * (Q

^{-1}* I

_{n}) * Q = (m + 1) * I

_{n}

(m + 1) * (Q

^{-1}) * Q = (m + 1) * I

_{n}

(m + 1) * I

_{n}= (m + 1) * I

_{n}.

Since they equal this is true by induction.

This is one of 6 bonus problems the linear algebra class were given. It's the first three people to turn in each of the problems gets the two points. However you only get one submission. So I thought I'd have someone else check my work to see if it is correct or not before I turn it in.

Thank you in advance!