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Diagonalization

Eigenspace

All eigenvectors of matrix \(A\) corresponding to \(\lambda_0\), together with the zero vector, form a linear space called an eigenspace of matrix \(A\), denoted as \(E(\lambda_0)\). It is the solution space of the homogeneous linear system:

\[ (\lambda_0 I-A)X=0 \]

For the eigenspace \(E(\lambda_i)=N(\lambda_i I-A)\), by the rank–nullity theorem:

\[ r(\lambda_i I-A)+\operatorname{dim} N(\lambda_i I-A)=n \]

Thus, the dimension of the eigenspace \(E(\lambda_i)\) is:

\[ \operatorname{dim} E(\lambda_i)=n-r(\lambda_i I-A) \]

This is also called the geometric multiplicity of \(\lambda_i\).

Invariant Subspace

When studying a linear transformation \(T\), it is often desirable to choose a basis for the space \(V\) such that the matrix of \(T\) with respect to this basis has the simplest possible form.

Let \(V\) be a linear space over a field \(F\), \(W\) a subspace of \(V\), and \(T\) a linear transformation on \(V\). If for every vector \(x\) in \(W\), \(T(x)\) is also in \(W\) (also said that the space is invariant or stable under the transformation), then \(W\) is called an invariant subspace of \(T\).

A space being invariant under a transformation does not mean the coordinates are truly "unchanged"; they might be stretched or otherwise deformed, but they remain within the space.

  • Any subspace of the linear space \(V\) is an invariant subspace under scalar multiplication.
  • For any linear transformation \(T\) on \(V\), the space \(V\) itself and the zero subspace are invariant subspaces of \(T\), called the trivial invariant subspaces.
  • The intersection and sum of invariant subspaces are also invariant subspaces.

Let \(W\) be an invariant subspace of the linear transformation \(T\). Considering only the action of \(T\) on the invariant subspace \(W\) yields a linear transformation on \(W\) itself, called the restriction of \(T\) to the subspace \(W\), denoted by \({T|}_W\).

For any linear transformation \(T\) on \(V\), the image space \(R(T)\) and the kernel space \(N(T)\) are invariant subspaces of \(T\). The meaning in these two cases is that the space \(V\), before and after the transformation, either compresses onto itself (image space) or compresses to \(0\) (kernel space).

For any linear transformation \(T\) on \(V\), the eigenspace of \(T\) is an invariant subspace of \(T\).

Primary Decomposition

According to the fundamental theorem of algebra, the minimal polynomial can be factored as:

\[ m_A(\lambda)={(\lambda-\lambda_1)}^{r_1}\cdots{(\lambda-\lambda_S)}^{r_S} \]

Consider substituting the variable \(\lambda\) in the minimal polynomial with the matrix \(A\), and then consider the kernel spaces of each factor, which constitute a series of invariant subspaces of matrix \(A\):

\[ W_i=N({(\lambda_i I-A)}^{r_i}) \]

Theorem: The dimension of this invariant subspace \(W_i\) is exactly the algebraic multiplicity of the eigenvalue \(\lambda_i\).

Recall that the algebraic multiplicity is the degree of each factor in the characteristic polynomial, and the geometric multiplicity is the dimension of the eigenspace \(E(\lambda_i)=N(\lambda_i I-A)\). This invariant subspace \(W_i\) and the eigenspace \(E(\lambda_i)\) are both kernel spaces of matrices, and the two matrices are related by the \(r_i\)-th power of the minimal polynomial. That is, the dimension of the eigenspace is the geometric multiplicity; the "eigenspace" is raised to the \(r_i\)-th power of the minimal polynomial to reach an "invariant subspace", whose dimension reaches the algebraic multiplicity of the characteristic polynomial.

This theorem is actually a corollary of the following primary decomposition theorem.

Denote the linear transformation \(T\) corresponding to matrix \(A\), and its restriction to each subspace \(W_i\) as \(T_i={T|}_{W_i}\). Then the minimal polynomial of \(T_i\) is \((x-\lambda_i)^{r_i}\).

Theorem: Let \(V\) be a linear space over a field \(F\), and \(T\) a linear transformation on \(V\). Then the space \(V\) can be decomposed into a direct sum of several invariant subspaces \(W_i\) with respect to the linear transformation \(T\).

\[ V=W_1\oplus W_2\oplus\cdots\oplus W_S \]

This means that the matrix of \(T\) under some basis is a block diagonal matrix:

\[ \operatorname{diag}\{A_1,A_2,\cdots,A_S\} \]

where \(A_i\) is the matrix of \(T_i\) under the corresponding basis.

This theorem shows that invariant subspaces can be used to simplify the matrix of a linear transformation.

Diagonalizable Matrix

For an \(n \times n\) matrix \(A\), if it is similar to a diagonal matrix, then \(A\) is called a diagonalizable matrix, or a simple matrix.

  • The sum, product, and inverse (if they exist) of diagonal matrices are still diagonal matrices, and the diagonal entries are their eigenvalues.
  • The matrix of a linear transformation \(T\) being diagonalizable is equivalent to the matrix of \(T\) under some basis being a diagonal matrix.

Theorem: Let the distinct eigenvalues of matrix \(A\) be \(\lambda_1,\cdots,\lambda_m\). Then the following statements are equivalent:

  • Matrix \(A\) is diagonalizable.
  • Matrix \(A\) has \(n\) linearly independent eigenvectors.
  • The following formula holds:
\[ \operatorname{dim} E(\lambda_1)+\cdots+\operatorname{dim} E(\lambda_m)=n \]

As noted earlier, the degree of an eigenvalue in the factorization of the characteristic polynomial is called its algebraic multiplicity, and the dimension of its eigenspace is called its geometric multiplicity. This theorem also shows that matrix \(A\) is diagonalizable if and only if for each eigenvalue \(\lambda\) of \(A\), its algebraic multiplicity equals its geometric multiplicity.

Corollary: If an \(n \times n\) matrix \(A\) has exactly \(n\) distinct eigenvalues, then it must be diagonalizable. The converse is not necessarily true.

Theorem: Matrix \(A\) is diagonalizable if and only if the minimal polynomial of \(A\) has no repeated roots.

Matrix similarity also preserves the linear dependence relations among eigenvectors.

Eigenvectors may not be real numbers, and it is entirely possible not to find \(n\) linearly independent eigenvectors.

For repeated eigenvalues, eigenvectors span a space. To describe this space, representatives need to be chosen from them. Usually, linearly independent representatives are chosen, and the number of representatives is the dimension of the space.

When selecting representatives, they are often orthogonalized and normalized. The final result is a set of orthonormal representatives.

Eigenvectors are not necessarily orthogonal; eigenvectors corresponding to different eigenvalues may not be orthogonal. Therefore, orthogonalization can only be performed on eigenvectors of repeated eigenvalues. However, normalization can be performed on any eigenvector.

Nilpotent Matrix

Let \(T\) be a linear transformation on space \(V\). If there exists a positive integer \(r\) such that \(T^r\) is the zero transformation, then \(T\) is called a nilpotent transformation on space \(V\).

A matrix \(N\) satisfying \(N^r=0\) for some positive integer \(r\) is called a nilpotent matrix.

Generally, we can further assume that \(r\) is the smallest positive integer such that \(T^r\) is the zero transformation, so the minimal polynomial of \(T\) is \(x^r\). Then there exists a vector \(\xi_0\) such that:

  • \[ T^r(\xi_0)=0 \]
  • \[ T^{r-1}(\xi_0)\neq 0 \]

Cyclic Subspace

Theorem: Let \(T\) be a linear transformation on space \(V\), and \(\xi\) a vector in \(V\). If there exists a positive integer \(s\) such that:

  • \[ T^s(\xi)=0 \]
  • \[ T^{s-1}(\xi)\neq 0 \]

Then the vectors \(\xi,T(\xi),\cdots,T^{s-1}(\xi)\) are linearly independent.

From this theorem, we can give a definition:

Let \(T\) be a linear transformation on space \(V\), and \(W\) a subspace of \(V\). If there exists a vector \(\xi_0\) and a positive integer \(r\) such that:

  • The vectors \(\xi_0,T(\xi_0),\cdots,T^{r-1}(\xi_0)\) form a basis for \(W\).

  • The following equality holds:

    \[ T^r(\xi_0)=0 \]

Then the subspace \(W\) is called a cyclic subspace with respect to \(T\), or simply a \(T\)-cyclic subspace. In this case, \(\xi_0\) is called a generating vector of the cyclic subspace \(W\), and the vectors \(\xi_0,T(\xi_0),\cdots,T^{r-1}(\xi_0)\) are called a cyclic basis of \(W\).

Clearly, a \(T\)-cyclic subspace \(W\) is invariant under \(T\), and for any vector \(\xi\) in the cyclic subspace \(W\), \(T^r(\xi)=0\), where \(r\) is the dimension of the cyclic subspace.

Nilpotent Jordan Block

If space \(W\) is a cyclic subspace of transformation \(T\), then the restriction \({T|}_W\) of \(T\) to \(W\) is a nilpotent transformation on \(W\), and the matrix of \({T|}_W\) with respect to the reverse-ordered cyclic basis \(T^{r-1}(\xi_0),T^{r-2}(\xi_0),\cdots,\xi_0\) of \(W\) is an \(r \times r\) upper triangular matrix of the following form:

\[ N_r=\begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0\\ 0 & 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & & \vdots& \vdots\\ 0 & 0 & 0 & \cdots & 0 & 1\\ 0 & 0 & 0 & \cdots & 0 & 0\\ \end{pmatrix} \]

The matrix \(N_r\) is called an \(r\)-th order nilpotent Jordan matrix, or an \(r\)-th order nilpotent Jordan block.

Let \(T\) be a nilpotent transformation on an \(n\)-dimensional space \(V\). The uniquely determined set of positive integers \(r_1\geq\cdots\geq r_S\) appearing in the decomposition of \(V\) into \(T\)-cyclic subspaces is called the invariant exponents of \(T\).

For an \(n \times n\) nilpotent matrix \(A\), \(A\) is similar to a matrix \(N\) of the above form, which also uniquely determines a sequence of positive integers \(r_1\geq\cdots\geq r_S\), called the invariant exponents of matrix \(A\).

Although a nilpotent matrix cannot be similar to a diagonal matrix, it can be similar to such a standard form. In the Jordan canonical form, the similarity diagonalization and the standard form of nilpotent matrices are combined to give the standard form that general matrices can achieve through similarity transformations.

Some Theorems

  1. Let \(T\) be a nilpotent transformation on space \(V\), and

    \[ h(x)=a_0+a_1x+\cdots+a_mx^m \]

    be a polynomial. Then the linear transformation \(h(T)\) is invertible if and only if \(a_0\neq 0\). When \(h(T)\) is invertible, its inverse is also a polynomial in \(T\).

  2. Let \(T\) be a nilpotent transformation on space \(V\), \(W\) an \(r\)-dimensional \(T\)-cyclic subspace, and \(\xi\) a vector in \(W\). If there exists an integer \(k\) such that

    \[ T^{r-k}(\xi)=0 \]

    then there exists a vector \(\eta\) in \(W\) such that

    \[ \xi=T^k(\eta) \]

  3. Let \(T\) be a nilpotent transformation on an \(n\)-dimensional space \(V\), \(x^r\) the minimal polynomial of \(T\), and \(W_1\) an \(r\)-dimensional \(T\)-cyclic subspace. Then there exists a complementary subspace \(W_2\) of \(W_1\) such that:

    \[ V=W_1\oplus W_2 \]

    and \(W_2\) is also invariant under \(T\).

  4. Let \(T\) be a nilpotent transformation on an \(n\)-dimensional space \(V\). Then \(V\) can be decomposed into a direct sum of \(T\)-cyclic subspaces:

    \[ V=W_1\oplus W_2\oplus\cdots\oplus W_S \]

  5. Every \(n \times n\) nilpotent matrix is similar to a matrix of the form:

    \[ N=\begin{pmatrix} N_{r_1} & & & 0\\ & N_{r_2} & & \\ & & \cdots & \\ 0 & & & N_{r_S}\\ \end{pmatrix} \]

    where each \(N_{r_i}\) is an \(r_i\)-th order nilpotent Jordan block.

  6. If the \(T\)-cyclic subspaces \(W_i\) are arranged in descending order of dimension \(r_1\geq\cdots\geq r_S\), then the decomposition of \(V\) into \(T\)-cyclic subspaces is uniquely determined by \(T\).

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