Finding the Base Matrix A from Matrix A100

[Anarza]

Homework Statement

If A¹⁰⁰ is some 3x3 matrix, find the base matrix A.

2. Relevant information
Eigenvalues, diagonalization, etc.

The Attempt at a Solution

So far, I've been finding the eigenvalues and diagonalizing the matrix via A = P^-1DP where D is the diagonal matrix and P is a matrix with A's eigenvalues. hoping to find the base matrix with powers of n. Any suggestions to guide me?

There was a similar discussion to this earlier, trying to reference that as well.

 

[RUber]

I think your method sounds good. What problem are you having with it?
If A = P^-1DP, then A² = P^-1DPP^-1DP = P^-1D²P, right?

 

[Ray Vickson]

Homework Statement

If A¹⁰⁰ is some 3x3 matrix, find the base matrix A.

2. Relevant information
Eigenvalues, diagonalization, etc.

The Attempt at a Solution

So far, I've been finding the eigenvalues and diagonalizing the matrix via A = P^-1DP where D is the diagonal matrix and P is a matrix with A's eigenvalues. hoping to find the base matrix with powers of n. Any suggestions to guide me?

There was a similar discussion to this earlier, trying to reference that as well.

Some matrices are not diagonalizable. What would you do in those cases?

 

[Anarza]

I think your method sounds good. What problem are you having with it?
If A = P^-1DP, then A² = P^-1DPP^-1DP = P^-1D²P, right?

After finding the diagonal, the P matrix, and the inverse of P, I'm setting them equal to Aⁿ (with the numeric digits in the diagonal also set to the nth power). The cross produce of these three matrices supposedly give me the power formula of the base matrix A.

Example: A 3x3 matrix of {{1,1,1},{2,2,2},{3,3,3}} turns out to be all the same digits to the nth power. I can see how this works moving forward, but it's not giving me the root matrix.

 

[Anarza]

Some matrices are not diagonalizable. What would you do in those cases?

Hmm, I'm not quite sure. I did a check to make sure the one I'm dealing with is, but I would be interested in knowing what to do in other cases as well. The book I'm using gives examples on how to go forward from A to A¹⁰⁰ via use of patterns, but those matricies are diagonalizable as well.

 

[RUber]

After finding the diagonal, the P matrix, and the inverse of P, I'm setting them equal to Aⁿ (with the numeric digits in the diagonal also set to the nth power). The cross produce of these three matrices supposedly give me the power formula of the base matrix A.

Example: A 3x3 matrix of {{1,1,1},{2,2,2},{3,3,3}} turns out to be all the same digits to the nth power. I can see how this works moving forward, but it's not giving me the root matrix.

I think your example may hint at Ray's question.
With a diagonalizable matrix A=P^-1DP, you have found that A¹⁰⁰=P^-1D¹⁰⁰P. And a diagonal matrix raised to a power is the same as each term in that matrix raised to the same power.
I question whether your example is diagonalizable, since your rows are all colinear.

 

[Ray Vickson]

Hmm, I'm not quite sure. I did a check to make sure the one I'm dealing with is, but I would be interested in knowing what to do in other cases as well. The book I'm using gives examples on how to go forward from A to A¹⁰⁰ via use of patterns, but those matricies are diagonalizable as well.

Here is an example:
$$A = \pmatrix{11/10&19/10&21/10\\ 1/10& 19/10& 11/10\\ -1/10&-9/10&-1/10}$$
The eigenvalues of $A$ are $1,1,9/10$, and there are only two eigenvectors; thus, $A$ is not diagonalizable. In fact, if
$$P = \pmatrix{1&0&1\\0&1&1\\-1&1&-1}, \; P^{-1} = \pmatrix{2&-1&1\\1&0&1\\-1&1&-1}$$
we have
$$A = P^{-1} J P \;\; \text{where} \;\; J = \pmatrix{1&1&0\\0&1&0\\0&0&9/10}$$
Here, $J$ is the Jordan canonical form of $A$.

We have $A^n = P^{-1} J^n P$, and $J^n$ is easily computed:
$$J^n = \pmatrix{1 & n & 0\\0 & 1 & 0\\0 & 0 & (9/10^n}$$

In general, if the Jordan canonical form of a matrix is
$$J = \pmatrix{a & 1 & 0\\0 & a & 0 \\ 0 & 0 & b}$$
then
$$J^n = \pmatrix{a^n & n a^{n-1} & 0 \\ 0 & a^n & 0 \\ 0 & 0 & b^n}$$

 

[Anarza]

Here is an example:
$$A = \pmatrix{11/10&19/10&21/10\\ 1/10& 19/10& 11/10\\ -1/10&-9/10&-1/10}$$
The eigenvalues of $A$ are $1,1,9/10$, and there are only two eigenvectors; thus, $A$ is not diagonalizable. In fact, if
$$P = \pmatrix{1&0&1\\0&1&1\\-1&1&-1}, \; P^{-1} = \pmatrix{2&-1&1\\1&0&1\\-1&1&-1}$$
we have
$$A = P^{-1} J P \;\; \text{where} \;\; J = \pmatrix{1&1&0\\0&1&0\\0&0&9/10}$$
Here, $J$ is the Jordan canonical form of $A$.

We have $A^n = P^{-1} J^n P$, and $J^n$ is easily computed:
$$J^n = \pmatrix{1 & n & 0\\0 & 1 & 0\\0 & 0 & (9/10^n}$$

In general, if the Jordan canonical form of a matrix is
$$J = \pmatrix{a & 1 & 0\\0 & a & 0 \\ 0 & 0 & b}$$
then
$$J^n = \pmatrix{a^n & n a^{n-1} & 0 \\ 0 & a^n & 0 \\ 0 & 0 & b^n}$$

Ah, that actually makes sense! Thank you for going through the details.