Undergrad Finding the Kernel of a Matrix Map

[WWGD]

Hi All,
I am trying to see if there is a "nice" ( relatively straightforward) way of finding the
solution/kernel of the map : $f(A)=A^n -Id$ , where A is an $k \times k$ matrix and $n$ is a positive
integer. Question: what is the kernel of this map? Cranking out matrix coefficients gets messy
and very slow as n grows. I can think of a solution using the Fundamental Theorem of Linear
Algebra relating Ortho spaces of rows, columns and row- and column- spaces. Is there some other
"reasonable" way of finding all solutions (clearly Id,-Id are solutions for n even and Id is a solution for n odd
)?

 

[StoneTemplePython]

some more details would be of interest. Since you mention orthogonality, I'm assuming your scalars are in $\mathbb R$ or $\mathbb C$.

What's the motivation here-- are you trying to calculate a (finite) von neumann series?

To confirm: is $Id$ the identity matrix $I$? If so, the kernel is really a question of whether or not $\mathbf A$ has an eigenvalue with magnitude 1 (and is an nth root of unity), and what the geometric multiplicities of associated eigenvectors are...

another thought: if you want to go the direct multiplication route there is the approach of first finding $\mathbf A^2$ then squaring that to get $\mathbf A^4$ then squaring that to get $\mathbf A^8$ then squaring that to get $\mathbf A^{16}$, and so on until you get 'close' to $\mathbf A^n$ then cleverly finishing off the residual... as I recall this cuts down multiplications to something like $O\big(log(n)\big)$,in a manner similar to bisection.

 

[WWGD]

some more details would be of interest. Since you mention orthogonality, I'm assuming your scalars are in $\mathbb R$ or $\mathbb C$.

What's the motivation here-- are you trying to calculate a (finite) von neumann series?

To confirm: is $Id$ the identity matrix $I$? If so, the kernel is really a question of whether or not $\mathbf A$ has an eigenvalue with magnitude 1 (and is an nth root of unity), and what the geometric multiplicities of associated eigenvectors are...

another thought: if you want to go the direct multiplication route there is the approach of first finding $\mathbf A^2$ then squaring that to get $\mathbf A^4$ then squaring that to get $\mathbf A^8$ then squaring that to get $\mathbf A^{16}$, and so on until you get 'close' to $\mathbf A^n$ then cleverly finishing off the residual... as I recall this cuts down multiplications to something like $O\big(log(n)\big)$,in a manner similar to bisection.

Yes, thanks, good points, yes, we would need a notion of orthogonality to make appeal to Fundamental Theorem, but I think it applies only to Real-valued matrices. Still, rotation by $2\pi/n$ is a solution , despite having no Real eigenvalues.

 

[Hawkeye18]

In the class of complex matrices all the solutions of the equation $A^n=I$ are given by $A=SDS^{-1}$, where $S$ is an invertible matrix, and $D$ is a diagonal matrix whose diagonal entries are $n$th roots of unity, $d_{j}^n=1$.

In the class of real matrices all the solutions are given by $A=SRS^{-1}$, where $S$ is a real invertible matrix, and $R$ is a "rotation" matrix, i.e. $R$ is a block diagonal matrix whose diagonal blocks can be either rotations through the angle $2\pi j/n$, $j=0, 1, 2, \ldots, n-1$ (these are $2\times 2$ blocks) or just real roots of unity ($1\times 1$ blocks). The real roots of unity are $\pm 1$ if $n$ is even, and $1$ if $n$ is odd.

A rotation through the angle $\alpha$ matrix is given by $$\left( \begin{array}{cc} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{array}\right).$$

 

[WWGD]

In the class of complex matrices all the solutions of the equation $A^n=I$ are given by $A=SDS^{-1}$, where $S$ is an invertible matrix, and $D$ is a diagonal matrix whose diagonal entries are $n$th roots of unity, $d_{j}^n=1$.

In the class of real matrices all the solutions are given by $A=SRS^{-1}$, where $S$ is a real invertible matrix, and $R$ is a "rotation" matrix, i.e. $R$ is a block diagonal matrix whose diagonal blocks can be either rotations through the angle $2\pi j/n$, $j=0, 1, 2, \ldots, n-1$ (these are $2\times 2$ blocks) or just real roots of unity ($1\times 1$ blocks). The real roots of unity are $\pm 1$ if $n$ is even, and $1$ if $n$ is odd.

A rotation through the angle $\alpha$ matrix is given by $$\left( \begin{array}{cc} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{array}\right).$$

Thanks, Hawkeye, are all permutation matrices then also rotation matrices? The antidiagonal matrix $A_{ij} ; a_{ii}=0 ; a_{ij}=-1 ; i \neq j$ satisfies $A^2=Id$ but it cannot be a rotation matrix since otherwise $sin \alpha =-sin \alpha =-1$. Did I make a mistake?EDIT: Or is this matrix conjugate to a rotation? EDIT2: I don't think it can be conjugate to a rotation matrix ; its determinant would still need to equal 1.

 

[Hawkeye18]

Thanks, Hawkeye, are all permutation matrices then also rotation matrices? The antidiagonal matrix $A_{ij} ; a_{ii}=0 ; a_{ij}=-1 ; i \neq j$ satisfies $A^2=Id$ but it cannot be a rotation matrix since otherwise $sin \alpha =-sin \alpha =-1$. Did I make a mistake?EDIT: Or is this matrix conjugate to a rotation? EDIT2: I don't think it can be conjugate to a rotation matrix ; its determinant would still need to equal 1.

I believe you are talking here about a $2\times 2$ matrix. Then your antidiagonal matrix is not a rotation matrix, but it is a matrix of form $SDS^{-1}$, where $$D=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).$$

 

[WWGD]

I believe you are talking here about a $2\times 2$ matrix. Then your antidiagonal matrix is not a rotation matrix, but it is a matrix of form $SDS^{-1}$, where $$D=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).$$

Right, but I thought all solutions to $A^n =Id$ consisted of rotations ( up to conjugation) , and the matrix I mentioned, antidiagonals =-1; 0 otherwise, is a solution (specifically, to $A^2=Id$), but not a rotation? But, I agree, if A is a rotation then its eigenvalues must be n-th roots of unity .

 

[Hawkeye18]

Let me elaborate. In the class of real matrices, all the solutions are given by $A=SRS^{-1}$, where $R$ is a "rotation" (note the quotes, that is not always a real rotation), and $S$ is invertible. "Rotation" here mean that $R$ is the block diagonal matrices, where each block is either a $2\times 2$ rotation matrix (through an allowed angle), or just $1\times 1$ block, with $n$th root of unity in it. I probably did not make it clear, but you do not have to have both types of blocks: your matrix can have only $2\times2$ blocks, or only $1\times1$ blocks, or any combination of these 2 (that adds up to the dimension).

The matrix $$R=D=\left(\begin{array}{cc} 1 & 0\\ 0& -1\end{array}\right)$$ is of that type, and it is not a rotation (that why I used quotes). But your matrix can be represented as $SDS^{-1}$, so your example does not contradict my statement.

 

[StoneTemplePython]

I think you two may be talking past each other.

Right, but I thought all solutions to $A^n =Id$ consisted of rotations

This is wrong. For example: $A^2 = I$ is an involution. Such a matrix does not need to be normal. (i.e. it is diagonalizable, but need not be unitarily diagonalizable.)

Permutation matrices are a special kind of orthogonal matrix (which is a special kind of unitary matrix). Rotation matrices are also orthogonal matrices. All orthogonal matrices are normal.

Another way to think about it is: Pick your favorite orthogonal matrix $P$, where $P^n = I$. Now do a non-unitary similarity transform, i.e. consider $S$ where $S^{-1} \neq S^*$.

Now take a look at $\big(S^{-1}PS\big)$. The spectrum of this is the same as for $P$ because the matrices are similar. But the $P$ is normal while the new matrix is not.

yet we still have

$P^n = I$

so

$\big(S^{-1}PS\big)^n =\big(S^{-1}P^n S\big) = \big(S^{-1}IS\big) = \big(S^{-1}S\big)= I$

 

[WWGD]

I think you two may be talking past each other.
This is wrong. For example: $A^2 = I$ is an involution. Such a matrix does not need to be normal. (i.e. it is diagonalizable, but need not be unitarily diagonalizable.)

Permutation matrices are a special kind of orthogonal matrix (which is a special kind of unitary matrix). Rotation matrices are also orthogonal matrices. All orthogonal matrices are normal.

Another way to think about it is: Pick your favorite orthogonal matrix $P$, where $P^n = I$. Now do a non-unitary similarity transform, i.e. consider $S$ where $S^{-1} \neq S^*$.

Now take a look at $\big(S^{-1}PS\big)$. The spectrum of this is the same as for $P$ because the matrices are similar. But the $P$ is normal while the new matrix is not.

yet we still have

$P^n = I$

so

$\big(S^{-1}PS\big)^n =\big(S^{-1}P^n S\big) = \big(S^{-1}IS\big) = \big(S^{-1}S\big)= I$

Thanks, I am aware not all solutions are rotations , by a straightforward counting argument, but I thought someone here had stated they are rotations, up to similarity.

 

[Hawkeye18]

Thanks, I am aware not all solutions are rotations , by a straightforward counting argument, but I thought someone here had stated they are rotations, up to similarity.

I used "rotation" in quotes for the lack of better term, it is not always a real rotation. Your example is up to similarity what I call "rotation" see post # 8 above for the details.

 

[WWGD]

I used "rotation" in quotes for the lack of better term, it is not always a real rotation. Your example is up to similarity what I call "rotation" see post # 8 above for the details.

Got it Hawk, thanks.