Jordan Normal Form
Jordan Decomposition
Let \(T\) be a linear transformation on an \(n\)-dimensional space \(V\). If the minimal polynomial of \(T\) is:
Then, by the primary decomposition theorem, the space \(V\) can be decomposed into a direct sum of subspaces:
where \(V_i=N\left({(A-\lambda_i I)}^{r_i}\right)\), with \(A\) being the matrix corresponding to \(T\). These subspaces are all invariant under \(T\).
Let the transformation \(T_i\) be the projection of \(V\) onto the subspace \(V_i\), i.e., construct polynomials \(u_i(T)\) such that:
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\[ T_i=u_i(T)\frac{m_A(T)}{{(T-\lambda_i T_e)}^{r_i}} \]
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\[ T_1+T_2+\cdots+T_k=T_e \]
Here, \(T_e\) denotes the identity transformation on \(V\). Then we have the following properties:
- The restriction of \(T_i\) to \(V_i\), \({T_i|}_{V_i}\), is the identity transformation on \(V_i\).
- If \(i \neq j\), the restriction of \(T_i\) to \(V_j\), \({T_i|}_{V_j}\), is the zero transformation on \(V_j\).
Thus, the transformation \(T_i\) maps every vector \(\xi\) in \(V\) to its component \(\xi_i\) in \(V_i\).
Construct the transformation:
Since each transformation \(T_i\) is a polynomial in \(T\), \(T_D\) is also a polynomial in \(T\). Therefore, each subspace \(V_i\) is invariant under \(T_D\).
From the above equations, the restriction of \(T_D\) to \(V_i\), \({T_D|}_{V_i}\), is a homothety (scaling) on \(V_i\) with scaling factor \(\lambda_i\). Hence, \(T_D\) is diagonalizable.
Construct:
Then \(T_N\) is also a polynomial in \(T\), so each subspace \(V_i\) is invariant under \(T_N\). For any vector \(\xi_i\) in \(V_i\), we have:
Let \(r\) be the maximum of all \(r_i\). Then for any vector \(\xi\) in \(V\), the \(r\)-th power of \(T_N\) maps \(\xi\) to the zero vector. Therefore, \(T_N\) is a nilpotent transformation.
Thus, every transformation \(T\) on \(V\) can be written as:
where \(T_D\) is diagonalizable and \(T_N\) is nilpotent. Since both \(T_D\) and \(T_N\) are polynomials in \(T\), they commute:
Theorem: Let \(T_1\) and \(T_2\) be two diagonalizable transformations on \(V\) such that \(T_1T_2=T_2T_1\). Then there exists a basis such that the matrices of \(T_1\) and \(T_2\) with respect to this basis are both diagonal.
Theorem: Let \(T\) be a linear transformation on an \(n\)-dimensional space \(V\). Then there exists a diagonalizable transformation \(T_D\) and a nilpotent transformation \(T_N\) such that:
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\[ T=T_D+T_N \]
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\[ T_DT_N=T_NT_D \]
Both are polynomials in \(T\), and they are uniquely determined by \(T\).
This theorem gives the decomposition of \(T\), called the Jordan decomposition of \(T\). \(T_D\) is called the diagonalizable part of \(T\), and \(T_N\) is called the nilpotent part of \(T\).
Similarly, we have the Jordan decomposition for matrices:
Theorem: Let \(A\) be an \(n \times n\) matrix. Then there exists a diagonalizable matrix \(D\) and a nilpotent matrix \(N\) such that:
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\[ A=D+N \]
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\[ DN=ND \]
Both are polynomials in \(A\), and they are uniquely determined by \(A\).
This theorem gives the decomposition of \(A\), called the Jordan decomposition of \(A\). \(D\) is called the diagonalizable part of \(A\), and \(N\) is called the nilpotent part of \(A\).
Lambda Matrices
The following section introduces a more general concept of matrices containing a variable parameter \(\lambda\), not just a table of numbers. This discussion is broader than that of matrices consisting solely of numbers.
For \(\lambda\)-matrices, the corresponding field becomes the field of rational functions in the variable \(\lambda\).
A matrix whose entries are polynomials in \(\lambda\) is called a \(\lambda\)-matrix, denoted by \(A(\lambda)\).
Since the polynomial field contains the number field, numerical matrices are special cases of \(\lambda\)-matrices. The characteristic matrix \(\lambda I-A\) of a numerical matrix \(A\) is a type of \(\lambda\)-matrix.
Elementary Operations on Lambda Matrices
For \(\lambda\)-matrices, we can similarly define addition, subtraction, multiplication, elementary operations, and rank. For square \(\lambda\)-matrices, we can similarly define determinant, minor, and algebraic cofactor.
For \(\lambda\)-matrices, elementary operations are mostly the same as for numerical matrices, except the row addition operation is modified as follows (using row operations as an example):
- Multiply a row by a polynomial \(\varphi(\lambda)\) in \(\lambda\) and add it to another row.
Note that the row scaling operation is not modified. This is because row addition does not change the determinant, while row scaling does. To preserve the rank properties over the polynomial field, determinants can only be changed within the number field.
The corresponding elementary matrices are also modified accordingly.
It is easy to see that the determinants of the three types of elementary matrices are all nonzero constants, hence they are all full-rank. Therefore, left or right multiplication by them does not change the rank of the \(\lambda\)-matrix.
If \(A(\lambda)\) can be transformed into \(B(\lambda)\) by a finite number of elementary operations, then \(A(\lambda)\) and \(B(\lambda)\) are said to be equivalent.
For \(\lambda\)-matrices, if they are equivalent, they have the same rank. The converse is not true, which distinguishes them from numerical matrices.
Smith Normal Form
Theorem: Let the rank of a \(\lambda\)-matrix be \(r\). Then \(A(\lambda)\) is always equivalent to:
where:
Each \(d_i(\lambda)\) is a monic polynomial, and adjacent polynomials satisfy the divisibility relation \(d_i(\lambda)|d_{i+1}(\lambda)\).
This standard form is called the Smith normal form, and \(d_i(\lambda)\) are called the invariant factors.
The specific method to find the Smith normal form is to perform elimination from the top-left to the bottom-right. Each time, the top-left element is the greatest common divisor of all remaining polynomials in the lower-right part, and using this element, the entire row and column are reduced to \(0\).
Theorem: The condition that \(A(\lambda)\) and \(B(\lambda)\) are equivalent is equivalent to the condition that \(A(\lambda)\) and \(B(\lambda)\) have exactly the same invariant factors.
Elementary Divisors
By the fundamental theorem of algebra, let the factorization of the invariant factors \(d_1(\lambda),d_2(\lambda),\cdots,d_m(\lambda)\) of \(A(\lambda)\) be:
where \(\lambda_1,\cdots,\lambda_S\) are distinct. Since:
the exponents \(e_{1j},e_{2j},\cdots,e_{mj}\) are non-decreasing, and the exponents in the last factor \(d_m(\lambda)\) are all nonzero.
All factors with positive exponents in the above expression are collectively called the elementary divisors of \(A(\lambda)\).
Note that elementary divisors are counted with multiplicity. If for some \(j\), the exponent \(e_{ij}\) appears multiple times, then the corresponding elementary divisor \({(\lambda-\lambda_j)}^{e_{ij}}\) should appear the same number of times.
The previous theorem states that \(A(\lambda)\) and \(B(\lambda)\) are equivalent if and only if they have exactly the same invariant factors. If the invariant factors are identical, naturally the elementary divisors are also identical, but the converse is not true. In fact, we have:
Theorem: \(A(\lambda)\) and \(B(\lambda)\) have exactly the same invariant factors if and only if they have exactly the same elementary divisors and the same rank.
Thus, "having exactly the same elementary divisors and the same rank" also becomes a condition for judging the equivalence of \(\lambda\)-matrices.
During elementary operations, one can also first transform \(A(\lambda)\) into a diagonal matrix, then find the elementary divisors and rank, and finally obtain the invariant factors to get the standard form. We have:
Theorem: Suppose \(A(\lambda)\) is equivalent to a diagonal matrix:
Then the powers \({(\lambda-\lambda_j)}^{e_{ij}}\) of all linear factors of \(f_1(\lambda),f_2(\lambda),\cdots,f_r(\lambda)\) constitute the elementary divisors of \(A(\lambda)\).
The specific method to construct invariant factors from elementary divisors and rank is: first classify the elementary divisors by their factors, arrange them in a table, place the factors of the same type in descending order of power in the same row, place the highest power of each type of factor in a column, pad the number of columns with \(1\)s to reach the rank \(r\), then the product of each column forms an invariant factor.
Application to Characteristic Matrices
If \(A\) and \(B\) are numerical matrices, then their characteristic matrices are \(\lambda\)-matrices. We have:
Theorem: The condition that numerical matrices \(A\) and \(B\) are similar is equivalent to the condition that their characteristic matrices \(\lambda I-A\) and \(\lambda I-B\) are equivalent.
Since the characteristic matrix \(\lambda I-A\) only contains \(n\) \(\lambda\)s on the main diagonal, its rank is \(n\). From the above reasoning, characteristic matrices of matrices of the same size always have the same rank. Thus, we have the equivalence:
Numerical matrices \(A\) and \(B\) are similar if and only if their characteristic matrices \(\lambda I-A\) and \(\lambda I-B\) have exactly the same elementary divisors.
For the characteristic matrix \(\lambda I-A\), elementary operations preserve equivalence, so they do not change the rank.
Observing the three types of elementary operations, since the only modified operation (row addition) does not change the determinant, in fact the three elementary operations only change the determinant by a constant factor, thus not altering the factorization and degree of the resulting polynomial.
Therefore, the determinant of the characteristic matrix \(\lambda I-A\) is a polynomial of degree \(n\). After transforming to Smith normal form via elementary operations, since the rank is \(n\), the determinant is the product of all invariant factors on the main diagonal, which is also equal to the product of all elementary divisors. Hence, the sum of the degrees of all elementary divisors of \(\lambda I-A\) equals \(n\).
Jordan Normal Form
The matrix
where all elements on the main diagonal are \(\lambda\), the elements immediately above the main diagonal are all \(1\), and all other positions are \(0\), is called a Jordan block belonging to \(\lambda\).
Clearly, a nilpotent Jordan matrix is a special case of a Jordan matrix, i.e., when \(\lambda = 0\).
Theorem: Let \(T\) be a transformation on an \(n\)-dimensional space \(V\), and let \(\lambda_1,\cdots,\lambda_k\) be all the distinct eigenvalues of \(T\). Then there exists a basis such that the matrix of \(T\) with respect to this basis has the form:
where
and \(J_{i1},\cdots,J_{is_i}\) are all Jordan blocks belonging to \(\lambda_i\).
This is because, first, from the minimal polynomial:
we have the primary decomposition:
where:
Here, \(A\) is the matrix corresponding to \(T\).
Let \(S_i\) be the restriction of \(T\) to \(V_i\), \({T|}_{V_i}\). Next, we attempt to perform Jordan decomposition for each \(S_i\).
Denote \(T_e\) as the identity transformation on \(V\). Different from the previous Jordan decomposition, let \(T_i\) be the nilpotent part in the Jordan decomposition of \(S_i\):
Thus, \(T_i\) is a nilpotent transformation on the subspace \(V_i\), and in fact it is also the restriction of \(T-\lambda_i T_e\) to \(V_i\), \({(T-\lambda_i T_e)|}_{V_i}\).
The subspace \(V_i\) can be decomposed into a direct sum of cyclic subspaces of the nilpotent transformation \(T_i\):
In each cyclic subspace \(W_{ij}\), take a cyclic basis and arrange it in reverse order to form a basis for \(V_i\). Then the matrix of \(T_i\) with respect to this basis has the form:
All \(N_{ij}\) are nilpotent Jordan blocks. Thus, for the chosen basis of \(V_i\), the matrix corresponding to \(S_i\) is:
Here, \(J_{i1},J_{i2},\cdots,J_{is_i}\) are all Jordan blocks belonging to \(\lambda_i\).
For each subspace \(V_i\), choose a basis as described above, and combine them to form a basis for \(V\). Then the matrix of \(T\) with respect to this basis takes the form specified in the theorem.
A matrix of the form:
where each \(J_i\) is a Jordan block, is called a Jordan normal form.
Theorem: Every \(n \times n\) matrix \(A\) is similar to a Jordan normal form. Except for the order of the Jordan blocks, the Jordan normal form similar to \(A\) is uniquely determined by \(A\).
Note that in the constructed matrix \(B_i\) above, the first term is a multiple of the identity matrix and naturally commutes with the second term. Therefore, the first term is the diagonalizable part of the Jordan decomposition of \(B_i\), and the second term is the nilpotent part of the Jordan decomposition of \(B_i\).
In the Jordan normal form corresponding to a matrix, the diagonal matrix formed by the elements on the main diagonal is the diagonalizable part of the Jordan normal form corresponding to the matrix. Replacing the main diagonal elements with \(0\) gives the nilpotent part of the Jordan normal form corresponding to the matrix.
Theorem: In the Jordan normal form of matrix \(A\), each Jordan block:
corresponds to an elementary divisor \({(\lambda-\lambda_i)}^{n_i}\) of the characteristic matrix \(\lambda I-A\). All elementary divisors of \(\lambda I-A\) correspond to all Jordan blocks in the Jordan normal form of \(A\).
This is because matrix \(A\) is similar to its Jordan normal form, so their characteristic matrices are also equivalent. Reducing the characteristic matrix of the Jordan normal form to Smith normal form reveals this.
By this theorem, using the elementary divisors of \(\lambda I-A\), we can write the Jordan normal form of \(A\).
A corollary is that matrix \(A\) is diagonalizable if and only if all elementary divisors of \(\lambda I-A\) are linear.
Frobenius Theorem
As mentioned above, the Smith normal form of an \(n \times n\) characteristic matrix has rank \(n\).
Theorem: Let the Smith normal form of the characteristic matrix \(\lambda I-A\) of matrix \(A\) be:
Then the last invariant factor \(d_n(\lambda)\) is exactly the minimal polynomial \(m_A(\lambda)\) of \(A\).
Corollary: The equivalent conditions for matrix \(A\) to be diagonalizable are:
- The minimal polynomial \(m_A(\lambda)\) has no repeated roots.
- The invariant factors of \(\lambda I-A\) have no repeated roots.
- All elementary divisors of \(\lambda I-A\) are linear.
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