Weyr canonical form

A basic Weyr matrix with eigenvalue ${\displaystyle \lambda }$ is an ${\displaystyle n\times n}$ matrix ${\displaystyle W}$ of the following form: There is a partition

${\displaystyle n_{1}+n_{2}+\cdots +n_{r}=n}$ of ${\displaystyle n}$ with ${\displaystyle n_{1}\geq n_{2}\geq \cdots \geq n_{r}\geq 1}$

such that, when ${\displaystyle W}$ is viewed as an ${\displaystyle r\times r}$ block matrix ${\displaystyle (W_{ij})}$, where the ${\displaystyle (i,j)}$ block ${\displaystyle W_{ij}}$ is an ${\displaystyle n_{i}\times n_{j}}$ matrix, the following three features are present:

1. The main diagonal blocks ${\displaystyle W_{ii}}$ are the ${\displaystyle n_{i}\times n_{i}}$ scalar matrices ${\displaystyle \lambda I}$ for ${\displaystyle i=1,\ldots ,r}$.
2. The first superdiagonal blocks ${\displaystyle W_{i,i+1}}$ are full column rank ${\displaystyle n_{i}\times n_{i+1}}$ matrices in reduced row-echelon form (that is, an identity matrix followed by zero rows) for ${\displaystyle i=1,\ldots ,r-1}$.
3. All other blocks of W are zero (that is, ${\displaystyle W_{ij}=0}$ when ${\displaystyle j\neq i,i+1}$).

In this case, we say that ${\displaystyle W}$ has Weyr structure ${\displaystyle (n_{1},n_{2},\ldots ,n_{r})}$.

The following is an example of a basic Weyr matrix.

In this matrix, ${\displaystyle n=9}$ and ${\displaystyle n_{1}=4,n_{2}=2,n_{3}=2,n_{4}=1}$. So ${\displaystyle W}$ has the Weyr structure ${\displaystyle (4,2,2,1)}$. Also,

and

Let ${\displaystyle W}$ be a square matrix and let ${\displaystyle \lambda _{1},\ldots ,\lambda _{k}}$ be the distinct eigenvalues of ${\displaystyle W}$. We say that ${\displaystyle W}$ is in Weyr form (or is a Weyr matrix) if ${\displaystyle W}$ has the following form:

where ${\displaystyle W_{i}}$ is a basic Weyr matrix with eigenvalue ${\displaystyle \lambda _{i}}$ for ${\displaystyle i=1,\ldots ,k}$.

The following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks. The basic Weyr matrix in the top-left corner has the structure (4,2,1) with eigenvalue 4, the middle block has structure (2,2,1,1) with eigenvalue -3 and the one in the lower-right corner has the structure (3, 2) with eigenvalue 0.

Let ${\displaystyle A}$ be a square matrix of order ${\displaystyle n}$ over an algebraically closed field and let the distinct eigenvalues of ${\displaystyle A}$ be ${\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{k}}$. The Jordan–Chevalley decomposition theorem states that ${\displaystyle A}$ is similar to a block diagonal matrix of the form

${\displaystyle A={\begin{bmatrix}\lambda _{1}I+N_{1}&&&\\&\lambda _{2}I+N_{2}&&\\&&\ddots &\\&&&\lambda _{k}I+N_{k}\\\end{bmatrix}}={\begin{bmatrix}\lambda _{1}I&&&\\&\lambda _{2}I&&\\&&\ddots &\\&&&\lambda _{k}I\\\end{bmatrix}}+{\begin{bmatrix}N_{1}&&&\\&N_{2}&&\\&&\ddots &\\&&&N_{k}\\\end{bmatrix}}=D+N}$

where ${\displaystyle D}$ is a diagonal matrix, ${\displaystyle N}$ is a nilpotent matrix, and ${\displaystyle [D,N]=0}$, justifying the reduction of ${\displaystyle N}$ into subblocks ${\displaystyle N_{i}}$. So the problem of reducing ${\displaystyle A}$ to the Weyr form reduces to the problem of reducing the nilpotent matrices ${\displaystyle N_{i}}$ to the Weyr form. This is leads to the generalized eigenspace decomposition theorem.

Given a nilpotent square matrix ${\displaystyle A}$ of order ${\displaystyle n}$ over an algebraically closed field ${\displaystyle F}$, the following algorithm produces an invertible matrix ${\displaystyle C}$ and a Weyr matrix ${\displaystyle W}$ such that ${\displaystyle W=C^{-1}AC}$.

Step 1

Let ${\displaystyle A_{1}=A}$

Step 2

1. Compute a basis for the null space of ${\displaystyle A_{1}}$.
2. Extend the basis for the null space of ${\displaystyle A_{1}}$ to a basis for the ${\displaystyle n}$-dimensional vector space ${\displaystyle F^{n}}$.
3. Form the matrix ${\displaystyle P_{1}}$ consisting of these basis vectors.
4. Compute ${\displaystyle P_{1}^{-1}A_{1}P_{1}={\begin{bmatrix}0&B_{2}\\0&A_{2}\end{bmatrix}}}$. ${\displaystyle A_{2}}$ is a square matrix of size ${\displaystyle n}$ − nullity ${\displaystyle (A_{1})}$.

Step 3

If ${\displaystyle A_{2}}$ is nonzero, repeat Step 2 on ${\displaystyle A_{2}}$.

1. Compute a basis for the null space of ${\displaystyle A_{2}}$.
2. Extend the basis for the null space of ${\displaystyle A_{2}}$ to a basis for the vector space having dimension ${\displaystyle n}$ − nullity ${\displaystyle (A_{1})}$.
3. Form the matrix ${\displaystyle P_{2}}$ consisting of these basis vectors.
4. Compute ${\displaystyle P_{2}^{-1}A_{2}P_{2}={\begin{bmatrix}0&B_{3}\\0&A_{3}\end{bmatrix}}}$. ${\displaystyle A_{2}}$ is a square matrix of size ${\displaystyle n}$ − nullity ${\displaystyle (A_{1})}$ − nullity${\displaystyle (A_{2})}$.

Step 4

Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices ${\displaystyle A_{1},A_{2},A_{3},\ldots }$ and associated invertible matrices ${\displaystyle P_{1},P_{2},P_{3},\ldots }$ until the first zero matrix ${\displaystyle A_{r}}$ is obtained.

Step 5

The Weyr structure of ${\displaystyle A}$ is ${\displaystyle (n_{1},n_{2},\ldots ,n_{r})}$ where ${\displaystyle n_{i}}$ = nullity${\displaystyle (A_{i})}$.

Step 6

1. Compute the matrix ${\displaystyle P=P_{1}{\begin{bmatrix}I&0\\0&P_{2}\end{bmatrix}}{\begin{bmatrix}I&0\\0&P_{3}\end{bmatrix}}\cdots {\begin{bmatrix}I&0\\0&P_{r}\end{bmatrix}}}$ (here the ${\displaystyle I}$'s are appropriately sized identity matrices).
2. Compute ${\displaystyle X=P^{-1}AP}$. ${\displaystyle X}$ is a matrix of the following form:

${\displaystyle X={\begin{bmatrix}0&X_{12}&X_{13}&\cdots &X_{1,r-1}&X_{1r}\\&0&X_{23}&\cdots &X_{2,r-1}&X_{2r}\\&&&\ddots &\\&&&\cdots &0&X_{r-1,r}\\&&&&&0\end{bmatrix}}}$.

Step 7

Use elementary row operations to find an invertible matrix ${\displaystyle Y_{r-1}}$ of appropriate size such that the product ${\displaystyle Y_{r-1}X_{r,r-1}}$ is a matrix of the form ${\displaystyle I_{r,r-1}={\begin{bmatrix}I\\O\end{bmatrix}}}$.

Step 8

Set ${\displaystyle Q_{1}=}$ diag ${\displaystyle (I,I,\ldots ,Y_{r-1}^{-1},I)}$ and compute ${\displaystyle Q_{1}^{-1}XQ_{1}}$. In this matrix, the ${\displaystyle (r,r-1)}$-block is ${\displaystyle I_{r,r-1}}$.

Step 9

Find a matrix ${\displaystyle R_{1}}$ formed as a product of elementary matrices such that ${\displaystyle R_{1}^{-1}Q_{1}^{-1}XQ_{1}R_{1}}$ is a matrix in which all the blocks above the block ${\displaystyle I_{r,r-1}}$ contain only ${\displaystyle 0}$'s.

Step 10

Repeat Steps 8 and 9 on column ${\displaystyle r-1}$ converting ${\displaystyle (r-1,r-2)}$-block to ${\displaystyle I_{r-1,r-2}}$ via conjugation by some invertible matrix ${\displaystyle Q_{2}}$. Use this block to clear out the blocks above, via conjugation by a product ${\displaystyle R_{2}}$ of elementary matrices.

Step 11

Repeat these processes on ${\displaystyle r-2,r-3,\ldots ,3,2}$ columns, using conjugations by ${\displaystyle Q_{3},R_{3},\ldots ,Q_{r-2},R_{r-2},Q_{r-1}}$. The resulting matrix ${\displaystyle W}$ is now in Weyr form.

Step 12

Let ${\displaystyle C=P_{1}{\text{diag}}(I,P_{2})\cdots {\text{diag}}(I,P_{r-1})Q_{1}R_{1}Q_{2}\cdots R_{r-2}Q_{r-1}}$. Then ${\displaystyle W=C^{-1}AC}$.