Lambda matrices, matrix polynomials, division of lamda matrices, remainder theorem, scalar matrix polynomials, Cayley-Hamilton theorem

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Lambda matrices, matrix polynomials, division of λ-matrices, remainder theorem, scalar matrix polynomials, Cayley-Hamilton theorem

Lambda matrix. A matrix whose elements are polynomials in the variable λ.

Let F[λ] be a polynomial domain consisting of the set of all polynomials in λ with coefficients in field F. A non-zero mxn matrix over F[λ]

is called a λ-matrix.

Example.

Matrix polynomial.. A matrix polynomial can take any of the following three forms:

where the coefficients A₀, A₁, .... , A_p are mxn matrices over a field F and the indeterminate λ is a number.

Example.

where the coefficients a₀, a₁, .... , a_p are numbers and the indeterminate C is a matrix.

where the coefficients A₀, A₁, .... , A_p are matrices and the indeterminate C is a matrix.

Representation of a λ-matrix as a matrix polynomial. Any mxn λ-matrix can be written as a matrix polynomial. Let “p” be the degree of the polynomial of highest degree found in A(λ). Then A(λ) can be written as the following matrix polynomial:

where A₀, A₁, .... , A_p are mxn matrices.

Example.

Singular and non-singular λ-matrices. The determinant of an n-square λ-matrix is a polynomial in λ and if this determinant vanishes identically we call the matrix singular. Otherwise it is called non-singular.

Proper and improper λ-matrices. An n-square λ-matrix A(λ) is called proper if the matrix A_pin the matrix polynomial

is non-singular. It is called improper if matrix A_p is singular.

Operations with λ-matrices. Consider the two n-square λ-matrices A(λ) and B(λ) and their matrix polynomial equivalents:

and

Equality of two λ-matrices. Two λ-matrices A(λ) and B(λ) are said to be equal if p = q and A_i = B_i (i = 0, 1, 2, ... , p) in their matrix polynomial representations.

Sum of two λ-matrices. The sum of A(λ) and B(λ), A(λ) + B(λ), is a λ-matrix C(λ) obtained by adding corresponding elements of A(λ) and B(λ).

The product A(λ) B(λ) is a λ-matrix or matrix polynomial of degree at most p + q. If either A(λ) or B(λ) is non-singular, the degree of A(λ) B(λ) and also B(λ) A(λ) is exactly p + q.

A λ-matrix and its matrix polynomial equivalent are identically equal and the equality is not disturbed by replacing λ with any scalar k of F. For example, putting λ = k in (4) yields

However, when λ is replaced by an n-square matrix C, two results can be obtained due to the fact that, in general, two n-square matrices do not commute. These two results correspond to

and

where A_R(C) is called the right functional value of A(λ) and A_L(C) is called the left functional value of A(λ).

Division of λ-matrices. Consider the matrix polynomials

and

If B(λ) is non-singular, then there exist unique matrix polynomials Q₁(λ), R₁(λ), Q₂(λ), and R₂(λ) where R₁(λ) and R₂(λ) are either zero or of degree less than that of B(λ), such that

If R₁(λ) = 0 , B(λ) is called a right divisor of A(λ).

If R₂(λ) = 0 , B(λ) is called a left divisor of A(λ).

Scalar matrix polynomials. Let

where the coefficients b₀, b₁, .... , b_q and the indeterminate λ are scalars from a number field F. A matrix polynomial B(λ) of the form

(where I_n is the identity matrix) is called a scalar matrix polynomial. A scalar matrix polynomial is a matrix polynomial whose coefficients are scalar matrices.

Example. The following is a scalar matrix polynomial:

Theorem 1. A scalar matrix polynomial B(λ) = b(λ)∙I_n commutes with every n-square matrix polynomial.

Theorem 2. If

and

then there exist unique matrix polynomials Q₁(λ) and R₁(λ) such that

and if R₁(λ) = 0 , b(λ)∙I_n divides A(λ).

Theorem 3. A matrix polynomial

of degree n is divisible by a scalar matrix polynomial B(λ) = b(λ)∙I_n if and only if every a_ij(λ) in A(λ) is divisible by b(λ).

The Remainder Theorem. Let A(λ) be an n-square λ-matrix over the polynomial domain F[λ]

and let B = [b_ij] be an n-square matrix over field F. Since λI - B is non-singular, we may write

and

where R₁ and R₂ are free of λ.

Theorem 4. If A(λ) is divided by λI - B, where B = [b_ij] is n-square, until remainders R₁ and R₂, free of λ, are obtained, then

and

(where A_R(B) and A_L(B) are the right and left functional values of A(λ)).

When A(λ) is a scalar matrix polynomial

the remainders are identical so that

Theorem 5. If a scalar matrix polynomial f(λ)∙I_n is divided by λI - B until a remainder R, free of λ, is obtained, then R = f(B).

Theorem 6. A scalar matrix polynomial f(λ)∙I_n is divisible by λI - B if and only if f(B) = 0.

Cayley-Hamilton Theorem. Every square matrix A = [a_ij] satisfies its characteristic equation Φ(λ) = 0.

Proof. Let A be a n-square matrix having characteristic matrix (λI - A) and characteristic equation Φ(λ) = |λI - A| = 0. A theorem on adjoints states that for any matrix A

A(adj A) = |A| I_n .

Applying this theorem to the characteristic matrix (λI - A) we get

(λI - A) ·adj (λI - A) = Φ(λ)·I .

Thus Φ(λ)·I is divisible by λI - A and, by Theorem 6, Φ(A) = 0.

References.

Ayres. Matrices (Schaum).