Weyr canonical form

A basic Weyr matrix with eigenvalue $\backslash lambda$ is an $n\backslash times\; n$ matrix $W$ of the following form: There is an integer partition

: $n\_1\; +\; n\_2+\; \backslash cdots\; +n\_r=n$ of $n$ with $n\_1\backslash ge\; n\_2\backslash ge\; \backslash cdots\; \backslash ge\; n\_r\backslash ge\; 1$

such that, when $W$ is viewed as an $r\; \backslash times\; r$ block matrix $(W\_\{ij\})$, where the $(i,\; j)$ block $W\_\{ij\}$ is an $n\_i\; \backslash times\; n\_j$ matrix, the following three features are present:

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

In this case, we say that $W$ has Weyr structure $(n\_1,\; n\_2,\; \backslash ldots\; ,\; n\_r)$.

The following is an example of a basic Weyr matrix.

In this matrix, $n=9$ and $n\_1=4,\; n\_2=2,\; n\_3=2,\; n\_4=1$. So $W$ has the Weyr structure $(4,2,2,1)$. Also,

and

Let $W$ be a square matrix and let $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_k$ be the distinct eigenvalues of $W$. We say that $W$ is in Weyr form (or is a Weyr matrix) if $W$ has the following form:

where $W\_i$ is a basic Weyr matrix with eigenvalue $\backslash lambda\_i$ for $i\; =\; 1,\; \backslash ldots\; ,\; k$.

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.

center

The Weyr canonical form $W=P^\{-1\}\; J\; P$ is related to the Jordan form $J$ by a simple permutation $P$ for each Weyr basic block as follows: The first index of each Weyr subblock forms the largest Jordan chain. After crossing out these rows and columns, the first index of each new subblock forms the second largest Jordan chain, and so forth.Sergeichuk, [https://arxiv.org/abs/0709.2485v1 "Canonical matrices for linear matrix problems"], Arxiv:0709.2485 [math.RT], 2007

That the Weyr form is a canonical form of a matrix is a consequence of the following result: Each square matrix $A$ over an algebraically closed field is similar to a Weyr matrix $W$ which is unique up to permutation of its basic blocks. The matrix $W$ is called the Weyr (canonical) form of $A$.

Let $A$ be a square matrix of order $n$ over an algebraically closed field and let the distinct eigenvalues of $A$ be $\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_k$. The Jordan–Chevalley decomposition theorem states that $A$ is similar to a block diagonal matrix of the form

A=

\begin{bmatrix}

\lambda_1I + N_1& & & \\

& \lambda_2I + N_2 & & \\

& & \ddots & \\

& & & \lambda_kI + N_k \\

\end{bmatrix}

=

\begin{bmatrix}

\lambda_1I & & & \\

& \lambda_2I & & \\

& & \ddots & \\

& & & \lambda_kI \\

\end{bmatrix}

+

\begin{bmatrix}

N_1& & & \\

& N_2 & & \\

& & \ddots & \\

& & & N_k \\

\end{bmatrix}

=

D+N

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

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

Step 1

Let $A\_1=A$

Step 2

1. Compute a basis for the null space of $A\_1$.
2. Extend the basis for the null space of $A\_1$ to a basis for the $n$-dimensional vector space $F^n$.
3. Form the matrix $P\_1$ consisting of these basis vectors.
4. Compute . $A\_2$ is a square matrix of size $n$ − nullity $(A\_1)$.

Step 3

If $A\_2$ is nonzero, repeat Step 2 on $A\_2$.

1. Compute a basis for the null space of $A\_2$.
2. Extend the basis for the null space of $A\_2$ to a basis for the vector space having dimension $n$ − nullity $(A\_1)$.
3. Form the matrix $P\_2$ consisting of these basis vectors.
4. Compute . $A\_2$ is a square matrix of size $n$ − nullity $(A\_1)$ − nullity$(A\_2)$.

Step 4

Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices $A\_1,\; A\_2,\; A\_3,\; \backslash ldots$ and associated invertible matrices $P\_1,\; P\_2,\; P\_3,\; \backslash ldots$ until the first zero matrix $A\_r$ is obtained.

Step 5

The Weyr structure of $A$ is $(n\_1,n\_2,\; \backslash ldots,\; n\_r)$ where $n\_i$ = nullity$(A\_i)$.

Step 6

1. Compute the matrix (here the $I$'s are appropriately sized identity matrices).
2. Compute $X=P^\{-1\}AP$. $X$ is a matrix of the following form:

:: .

Step 7

Use elementary row operations to find an invertible matrix $Y\_\{r-1\}$ of appropriate size such that the product $Y\_\{r-1\}X\_\{r,r-1\}$ is a matrix of the form $I\_\{r,r-1\}=\; \backslash begin\{bmatrix\}\; I\; \backslash \backslash \; O\; \backslash end\{bmatrix\}$.

Step 8

Set $Q\_1=$ diag $(I,I,\; \backslash ldots,\; Y\_\{r-1\}^\{-1\},\; I)$ and compute $Q\_1^\{-1\}XQ\_1$. In this matrix, the $(r,r-1)$-block is $I\_\{r,r-1\}$.

Step 9

Find a matrix $R\_1$ formed as a product of elementary matrices such that $R\_1^\{-1\}\; Q\_1^\{-1\}XQ\_1R\_1$ is a matrix in which all the blocks above the block $I\_\{r,r-1\}$ contain only $0$'s.

Step 10

Repeat Steps 8 and 9 on column $r-1$ converting $(r-1,\; r-2)$-block to $I\_\{r-1,r-2\}$ via conjugation by some invertible matrix $Q\_2$. Use this block to clear out the blocks above, via conjugation by a product $R\_2$ of elementary matrices.

Step 11

Repeat these processes on $r-2,r-3,\backslash ldots\; ,\; 3,\; 2$ columns, using conjugations by $Q\_3,\; R\_3,\backslash ldots\; ,\; Q\_\{r-2\},\; R\_\{r-2\},\; Q\_\{r-1\}$. The resulting matrix $W$ is now in Weyr form.

Step 12

Let $C\; =\; P\_1\; \backslash text\{diag\}\; (I,\; P\_2)\; \backslash cdots\; \backslash text\{diag\}(I,\; P\_\{r-1\})Q\_1R\_1Q\_2\backslash cdots\; R\_\{r-2\}Q\_\{r-1\}$. Then $W\; =\; C^\{-1\}AC$.

Some well-known applications of the Weyr form are listed below:

1. The Weyr form can be used to simplify the proof of Gerstenhaber’s Theorem which asserts that the subalgebra generated by two commuting $n\; \backslash times\; n$ matrices has dimension at most $n$.
2. A set of finite matrices is said to be approximately simultaneously diagonalizable if they can be perturbed to simultaneously diagonalizable matrices. The Weyr form is used to prove approximate simultaneous diagonalizability of various classes of matrices. The approximate simultaneous diagonalizability property has applications in the study of phylogenetic invariants in biomathematics.
3. The Weyr form can be used to simplify the proofs of the irreducibility of the variety of all k-tuples of commuting complex matrices.

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Category:Linear algebra

Category:Matrix theory

Category:Matrix normal forms

Category:Matrix decompositions