# Exercises

Find square roots of the positive definite 2 × 2 matrix with complex coefficientsPaul A. Samuelson A Method of Determining Explicitly the Coefficients of the Characteristic Equation, The Annals of Mathematical Statistics, Volume 13, Number 4 (1942), 424-429.

The characteristic polynomial of a matrix**A**may be computed in the Wolfram Language as

`CharacteristicPolynomial[A, lambda]`

.
*Mathematica*, we find its chracteristic polynomial

CharacteristicPolynomial[A, x]

**A**is not diagonalizable because its minimal polynomial equals to the characteristic polynomial.

*T*be a linear transformation on a vector space

*V*. A subspace

*U*of

*V*is called a

*T*-

**invariant subspace**of

*V*if \( T(U) \subseteq U , \) that is, if \( T({\bf v}) \in U \) for all \( {\bf v} \in U . \)

*T*is a linear operator on a vector space

*V*. Then the following subspaces of

*V*are

*T*-invariant.

- The zero space {
**0**}. - The original space
*V*. - The range of
*T*; that is, if*T*is defined by a matrix**A**in some basis, then the column space is**A**-invariant. - The kernel (= null space) of
*T*. - The eigenspace
*E*_{λ}for any eigenvalue λ of*T*.

*T*be a linear operator on a vector space

*V*, and let

**x**be z nonzero element of

*V*. The subspace

*T*-

**cyclic subspace of**

*V*generated by x
A cyclic subspace is a "smallest" *T*-invariant subspace of the vector space *V* containing vector **x**.
We will use cyclic subspaces to establish the Cayley--Hamilton theorem, which is our main objective.

*x*

^{2}is span \( \left\{ x^2 , 2x, 2 \right\} = P_2 (\mathbb{R}) . \) ■

Let **A** be a square n × n matrix and \( p(\lambda ) = a_s \lambda^s + a_{s-1} \lambda^{s-1} + \cdots
+ a_1 \lambda + a_0 \) be a polynomial in variable λ of degree *s*. A polynomial for which
\( p({\bf A} ) = {\bf 0} \) is called the **annihilating poilynomial** for the matrix
**A** or it is said that *p(λ)* is an annihilator for matrix **A**.

An annihilating polynomial for a given square matrix is not unique and it could be multiplied by any polynomial.

*Mathematica*to verify that the characteristic polynomial \( \chi (\lambda ) = \left( \lambda -1 \right)^3 \left( \lambda +1 \right) \) annihilates the matrix:

p[x_] = CharacteristicPolynomial[A , x]

Theorem: Let *T* be a linear opeartor on a finite-dimensional vector space
*V*, and let *U* be a *T*-invariant subspace of *V*. Then the characteristic polynomial of
*T*_{U} divides the characteristic polynomial of *T*.

*U*and extend it to the basis \( \beta = \left\{ {\bf v}_1 , {\bf v}_2 , \ldots , {\bf v}_k , \ldots , {\bf v}_n \right\} \) for

*V*/ Let

**A**be the matrix representation of the operator

*T*in this basis and

**B**

_{1}be the matrix for operator

*T*

_{U}. Then

**A**will be the block matrix:

**B**

_{2}and

**B**

_{3}have appropriate sizes. Let χ(λ) be the characteristic polynomial of

*T*and

*g*(λ) bethe characteristic polynomial of

*T*

_{U}. Then

*g*(λ) divides χ(λ). ■

Theorem: Let *T* be a linear opeartor on a finite-dimensional vector space
*V*, and let *U* denote the *T*-cyclic subspace of *V* generated by a nonzero vector
\( {\bf v} \in V . \) Let \( k = \mbox{dim}(U) . \) Then
\( \left\{ {\bf v} , T\,{\bf v} , \ldots , T^{k-1} {\bf v} \right\} \) is the basis for
*U* so there exist scalars *a*_{0}, *a*_{1},...,*a*_{k-1} such that

*T*

_{U}.

*j*be the largest positive integer for which

*j*exist because

*V*is finite-dimensional space. Let

*Z*= span(β). Then β is a basis for

*Z*. Furthemore, \( T^j {\bf v} \in Z . \) We use this information to show that

*Z*is a

*T*-invariant subspace of

*V*. Let \( {\bf u} \in Z . \) Since

**u**is a linear combination of the elements of β, there exist scalars

*b*

_{0},

*b*

_{1}, ... ,

*b*

_{j-1}such that

*T*

**u**is a linear combination of elements of

*Z*, and hence belongs to

*Z*. So

*Z*is

*T*-invariant. Furthermore, \( {\bf v} \in Z . \) We know that

*U*is the smallest

*T*-invariant subspace of

*V*that contains

**v**, and therefore, \( U \subseteq Z . \) Clearly \( Z \subseteq U , \) and so we conclude that

*Z*=

*U*. It follows that β is a basis for

*U*and therefore dim(

*U*) =

*j*. Thus

*j*=

*k*.

Now view β as an ordered basis for *U*. Let *a*_{0}, *a*_{1}, ... , *a*_{k-1}
be the scalars such that

*T*

_{U}in basis β is

*f*(λ) is the characteristic polynomial for

*T*

_{U}. ■

Theorem (Cayley--Hamilton):
For a given n×n matrix **A** and **I**_{n} the n×n identity matrix, the characteristic
polynomial of **A** is defined as
\( \chi (\lambda ) = \det \left( \lambda {\bf I} - {\bf A} \right) , \)
where det is the determinant operation and λ is a scalar (real or complex). Then the characteristic polynomial
annihilates the matrix **A**: \( \chi ({\bf A} ) = {\bf 0} . \)
■

**v**from

*V*. It is obvious when

**v**=

**0**because \( \chi ({\bf A} ) \) is linear. Now suppose that \( {\bf v} \ne {\bf 0} . \) Let

*U*be the

**A**-cyclic subspace generated by

**v**, and suppose that dim (

*U*) =

*k*. Then, for some scalars \( a_0 , a_1 , \ldots , a_k \)

*U*is the

**A**-cyclic subspace. Hence \( g(\lambda ) = a_0 + a_1 \lambda + \cdots + a_{k-1} \lambda^{k-1} + \lambda^k \) is the characteristic polynomial of

*T*

_{U}. For the vector

**v**, we have

*g*(λ) divides χ(λ). ■

Arthur Cayley | William Hamilton |

In 1838 Arthur began his studies at Trinity College, Cambridge, having George Peacock as tutor in his first year. He was coached by William Hopkins who encouraged him to read papers by continental mathematicians. His favourite mathematical topics were linear transformations and analytical geometry and while still an undergraduate he had three papers published in the newly founded Cambridge Mathematical Journal edited by Duncan Gregory. Cayley graduated as Senior Wrangler in 1842 and won the first Smith's prize. After the examinations, Cayley and his friend Edmund Venables led a reading party of undergraduates to Aberfeldy in Scotland.

The Cambridge fellowship had a limited tenure, since Cayley was not prepared to take Holy Orders, so he had to find a profession. He chose law and began training in April 1846. While still training to be a lawyer Cayley went to Dublin to hear William Rowan Hamilton lecture on quaternions. He sat next to George Salmon during these lectures and the two were to exchange mathematical ideas over many years. Cayley was a good friend of Hamilton's although the two disagreed as to the importance of the quaternions in the study of geometry. Another of Cayley's friends was James Joseph Sylvester who was also in the legal profession. The two both worked at the courts of Lincoln's Inn in London and they discussed deep mathematical questions during their working day. Others at Lincoln's Inn who were active mathematicians included Hugh Blackburn.

Sir William Rowan Hamilton (1805--1865) was an Irish mathematician, astronomer, and mathematical physicist, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques. His best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions.

The Cayley--Hamilton theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general case was first proved by the German mathematician Ferdinand Georg Frobenius (1849--1917) in 1878.