# What Are the Minimal Polynomials of Matrix Powers?

• MHB
• evinda
In summary, the minimal polynomial of a matrix $A$ is the polynomial that minimizes the product of the matrix's elements. The minimal polynomial of the matrix $A^2$ is the polynomial that minimizes the product of the matrix's elements multiplied by the matrix's determinant, and the minimal polynomial of the matrix $A^3$ is the polynomial that minimizes the product of the matrix's elements multiplied by the matrix's stretches.
evinda
Gold Member
MHB
Hello! (Wave)

If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$.

($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$)

Is there a relation that connects the minimal polynomial of a matrix $B$ with the minimal polynomial of the powers of $B$ ? (Thinking)

evinda said:
Hello! (Wave)

If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$.

($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$)

Is there a relation that connects the minimal polynomial of a matrix $B$ with the minimal polynomial of the powers of $B$ ? (Thinking)

Hey evinda! (Wave)

Don't we have that $\lambda$ is an eigenvalue of $A$ iff $\lambda$ is a root of the minimal polynomial?

Now suppose $\lambda$ is an eigenvalue of $A$ with eigenvector $\mathbf v$.
Then what can we say about $A^2\mathbf v$? (Wondering)

I like Serena said:
Hey evinda! (Wave)

Don't we have that $\lambda$ is an eigenvalue of $A$ iff $\lambda$ is a root of the minimal polynomial?

(Nod)
I like Serena said:
Now suppose $\lambda$ is an eigenvalue of $A$ with eigenvector $\mathbf v$.
Then what can we say about $A^2\mathbf v$? (Wondering)
$A^2\mathbf v= A(A \mathbf v)=A(\lambda \mathbf v)=\lambda(A \mathbf v)=\lambda(\lambda \mathbf{v})=\lambda^2 \mathbf v$

and

$A^3 \mathbf v=A(A^2 \mathbf v)=A(\lambda^2 \mathbf v)=\lambda^2(A \mathbf v)=\lambda^3 \mathbf v$.

So the eigenvalues of $A^2$ are $\lambda^2$ and the eigenvalues of $A^3$ are $\lambda^3$.

In our case, the eigenvalues of $A$ are $\pm i$ and $\pm 1$.

Thus the eigenvalues of $A^2$ are $-1,1$ and the eigenvalues of $A^3$ are $-i, i, -1,1$.

So the minimal polynomials of $A^2$ and $A^3$ are $(x+1)(x-1)=x^2-1$ and $(x+i)(x-i)(x+1)(x-1)=(x^2+1)(x^2-1)$,respectively.

Right? (Thinking)

evinda said:
(Nod)

$A^2\mathbf v= A(A \mathbf v)=A(\lambda \mathbf v)=\lambda(A \mathbf v)=\lambda(\lambda \mathbf{v})=\lambda^2 \mathbf v$

and

$A^3 \mathbf v=A(A^2 \mathbf v)=A(\lambda^2 \mathbf v)=\lambda^2(A \mathbf v)=\lambda^3 \mathbf v$.

So the eigenvalues of $A^2$ are $\lambda^2$ and the eigenvalues of $A^3$ are $\lambda^3$.

In our case, the eigenvalues of $A$ are $\pm i$ and $\pm 1$.

Thus the eigenvalues of $A^2$ are $-1,1$ and the eigenvalues of $A^3$ are $-i, i, -1,1$.

So the minimal polynomials of $A^2$ and $A^3$ are $(x+1)(x-1)=x^2-1$ and $(x+i)(x-i)(x+1)(x-1)=(x^2+1)(x^2-1)$,respectively.

Right?

Right! (Nod)

I like Serena said:
Right! (Nod)

Nice... Thanks a lot! (Happy)

## What is the minimal polynomial of a matrix?

The minimal polynomial of a matrix is the monic polynomial of lowest degree that has the matrix as a root. In other words, it is the smallest polynomial that satisfies the equation P(A) = 0, where A is the given matrix.

## How is the minimal polynomial of a matrix related to its eigenvalues?

The minimal polynomial of a matrix is closely related to its eigenvalues. In fact, the eigenvalues of a matrix are the roots of its minimal polynomial. This means that the minimal polynomial can give us information about the eigenvalues of a matrix.

## Can a matrix have multiple minimal polynomials?

Yes, it is possible for a matrix to have multiple minimal polynomials. This can occur when the matrix has repeated eigenvalues, as there may be multiple polynomials of the same degree that have the matrix as a root.

## How can the minimal polynomial of a matrix be calculated?

The minimal polynomial of a matrix can be calculated using various methods, such as the characteristic polynomial or the Cayley-Hamilton theorem. There are also algorithms, such as the Berlekamp-Massey algorithm, that can be used to find the minimal polynomial of a matrix.

## What is the significance of the minimal polynomial in linear algebra?

The minimal polynomial plays a crucial role in linear algebra, as it can be used to determine important properties of a matrix, such as its diagonalizability and its Jordan canonical form. It also has applications in areas such as control theory and differential equations.

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