Download Characteristic Functions and Eigenvalues of Matrices: An Introduction and more Slides Algebra in PDF only on Docsity!
Matrices and Transformations
Anthony J. Pettofrezzo[1]
No Institute Given
1 Eigenvalues and Eigenvectors
1.1 Characteristic Functions
Associated with each square matrix A = ((aij )) of order n is a function
f (λ) = |A − λI| =
a 11 − λ a 12... a 1 n a 21 a 22 − λ... a 2 n
............ an 1 an 2... ann − λ
called the characteristic function of A. The equation
f (λ) = |A − λI| = 0 (2) can be expressed in the polynomial form
c 0 λn^ + c 1 λn−^1 + · · · + cn− 1 λ + cn = 0 (3) and is called the characteristic equation of matrix A.
Example 1. Find the characteristic equation of matrix A where
A =
The characteristic equation of A is ∣ ∣ ∣ ∣ ∣∣
1 − λ 2 0 2 2 − λ 2 0 2 3 − λ
that is,
(1 − λ)
1 − λ 2 2 2 − λ
∣ −^2
0 3 − λ
(1 − λ)(λ^2 − 5 λ + 2) − 2(6 − 2 λ) = 0
λ^3 − 6 λ^2 + 3λ + 10 = 0
In some instances the task of expressing the characteristic equation of a matrix in polynomial form may be simpli ed considerably by introducing the concept of the trace of a matrix. The sum of the diagonal elements of a matrix A is denoted by tr(A) For example, the trace of matrix A in Example 1 is 1+2+3; that is, 6. Let t 1 = tr(A), t 2 = tr(A^2 ),... , tn = tr(An). It can be shown that the coe cients of the characteristic equation are given by the equations:
c 0 = 1 , c 0 = −t 1 , c 0 = − 12 (c 1 t 1 + t 2 ), c 0 = − 13 (c 2 t 1 + c 1 t 2 + t 3 ), · · · · · · c 0 = − (^1) n (cn− 1 t 1 + cn− 2 t 2 + · · · + c 1 tn− 1 + tn).
Equation 4 make it possible to calculate the coe cients of the characteristic equation of a matrix A by assuming the diagonal elements of the matrices of the form An. This numerical process is easily programmed on a large-scale digital computer, or for small values of n may be computed manually without di culty. The n roots λ 1 , λ 2 ,... , λn of the characteristic equation 3 of a matrix A are called the eigenvalues of A. The trace of a matrix A of order n is equal to the sum of the n eigenvalues of A. Many applications of matrix algebra in mathematics, physics, and engineer- ing involve the concept of a set of nonzero vectors being mapped onto the zero vector by means of the matrix A − λiI, where λi, is an eigenvalue of matrix A. Any nonzero column vector, denoted by Xi, such that
(A − λiI)Xi = 0 (5) is called eigenvector of matrix A. It is guaranteed that at least one eigenvec- tor exists for each λi since equation 5 represents a system of n linear homogeneous equations which has a nontrivial solution Xi 6 = 0 if and only if |A−λiI| = 0; that is, if and only if λi is an eigenvalue of A. Furthermore, note that any nonzero scalar multiple of an eigenvector associated with an eigenvalue is also an eigen- vector associated with that eigenvalue. The eigenvalues of a matrix are also called the proper values, the latent values, and the characteristic values of the matrix. The eigenvectors of a matrix are also called the proper vectors, the latent vectors, and the char- acteristic vectors of the matrix.
Example 2. Determine a set of eigenvectors of the matrix A =
Associated with λ 1 = 1 are the eigenvectors
x 1 x 2
)T
for which
(A − I)
x 1 x 2
)T
Fig. 1.
Similarly, every eigenvector associated with λ 2 is of the form
0 k
)T
, where
k is any nonzero scalar. The set of vectors of the form
0 k
)T
is such that
A
0 k
)T
= λ 1
0 k
)T
that is, ( 3 0 0 2
k
k
Hence the set of eigenvectors associated with λ 2 = 2 is mapped onto itself under the transformation represented by A, and the image of each eigenvector is a xed scalar multiple of the eigenvalue. The xed scalar multiple is λ 2 ; that is, 2.
Note that the sets of vectors of the forms
k 0
)T
and
0 k
)T
lie along the x -axis and y-axis, respectively 1.2. Under the magni cation of the plane repre- sented by the matrix
A =
( the^ one-dimensional vector spaces^ containing the sets of vectors of the forms k 0
)T
and
0 k
)T
are mapped onto themselves, respectively, and are called invariant vector spaces. The invariant vector spaces help characterize or de- scribe a particular transformation of the plane.
1.3 Some Theorems
In this section several theorems concerning the eigenvalues and eigenvectors of matrices in general and of symmetric matrices in particular will be proved.
These theorems are important for an understanding of the remaining sections of this text. Notice that in Example 2 the eigenvector associated with the distinct eigen- values of matrix A are linearly independent; that is,
k 1
implies k 1 = k 2 = 0. This is not a coincidence. The following theorem states a su cient condition for eigenvector associated with the eigenvalues of a matrix to be linearly independent.
Theorem 1. If the eigenvalues of a matrix are distinct, then the associated eigenvectors are linearly independent.
Proof. Let A be a square matrix of order n with distinct eigenvalues λ 1 , λ 2 ,... , λn and associated eigenvectors X 1 , X 2 ,... , Xn, respectively. Assume that the set of eigenvectors are linearly dependent. Then there exists scalars k 1 , k 2 ,... , kn, not all zero, such that
k 1 X 1 + k 2 X 2 + · · · + knXn = 0 (6) Consider premultiplying both sides of 6 by
(A − λ 2 I)(A − λ 3 I) · · · (A − knI) By use of equation 5, obtain
k 1 (A − λ 2 I)(A − λ 3 I) · · · (A − knI)X 1 = 0 (7) Since (A − λiI)Xi = 0, then AX 1 = λ 1 X 1. Hence, equation 7 may be written as
k 1 (λ 1 − λ 2 )(λ 1 − λ 3 ) · · · (λ 1 − λn)X 1 = 0 which implies k 1 = 0. Similarly, it can be shown that k 1 = k 2 = · · · = kn = 0, which is contrary to the hypothesis. Therefore, the set of eigenvectors are linearly independent. It should be noted that if the eigenvalues of a matrix are not distinct, the associated eigenvectors may or may not be linearly independent. For example, consider the matrices
A =
and B =
Both matrices have λ 1 = λ 2 = 3; that is, an eigenvalue of multiplicity two.
Any nonzero vector of the form
x 1 x 2
)T
is an eigenvector of A for λ 1 and λ 2. Hence, it is possible to choose any two linearly independent vectors such as ( 1 0
)T
and
)T
as eigenvectors of A that are associated with λ 1 and λ 2 ,
respectively. Only a vector of the form
x 1 0
)T
is, however, an eigenvector of B for λ 1 and λ 2. Any two vectors of this form are linearly dependent; that is, one is a linear function of the other.
Before presenting the next theorem it is necessary to consider the following de nition: two complex eigenvectors X 1 and X 2 are de ned as orthogonal if X 1 ∗ X 2 = 0. For example, if X 1 =
−i 2
)T
and X 2 =
2 i 1
)T
, then X 1 ∗ X 2 = ( i 2
2 i 1
)T
= 0. Hence, X 1 and X 2 are orthogonal.
Theorem 4. If A is a Hermitian matrix, then the eigenvectors of A associated with distinct eigenvalues are mutually orthogonal vectors.
Proof. Let A be a Hermitian matrix, and let X 1 and X 2 be eigenvectors associ- ated with any two distinct eigenvalues λ 1 and λ 2 , respectively. Then
(A − λ 1 I)X 1 = 0 and (A − λ 2 I)X 2 = 0
...
Theorem 5. If A is a real symmetric matrix, then the eigenvectors of A asso- ciated with distinct eigenvalues are mutually orthogonal vectors.
Inverse of a Matrix This section will be concerned with the problem of nding a multiplicative inverse, if it exists, for any given square matrix. A left mul- tiplicative inverse of a matrix A is a matrix B such that BA = I; a right multiplicative inverse of a matrix A is a matrix C such that AC = I. If a left and a right multiplicative inverse of a matrix A are equal, the the left (right) inverse is called, simply, a multiplicative inverse of A and is denoted by A−^1.
Theorem 6. A left multiplicative inverse of a square matrix A is a multiplica- tive inverse of A.
Proof. Suppose BA = I, then ....
Theorem 7. A right multiplicative inverse of a square matrix A is a multiplica- tive inverse of A.
Theorem 8. The multiplicative inverse, if it exists, of a square matrix A is unique
Proof. Let A−^1 and B be any of two multiplicative inverses of the square matrix A. Since A−^1 A = I and BA = I, then .... Not every square matrix has a multiplicative inverse. In fact, the necessary and su cient condition for the multiplicative inverse of a matrix A to exist is that det A 6 = 0. A square matrix A is said to be nonsingular if det A 6 = 0, and singular if det A = 0. It should be mentioned that if A is not a square matrix, then it is possible for A to have a left or a right multiplicative inverse, but not both.
1.4 Diagonalization of Matrices
It has been noted that an eigenvector Xi such that (A − λiI)Xi = 0, for i = 1 , 2 ,... , n, may be associated with each eigenvalue λi. This relationship may be expressed in the alternate form
AXi = λiXi f or i = 1, 2 ,... , n (8)
If a square matrix of order n whose columns are eigenvectors Xi of A is constructed and denoted by X, then the equations of 8 may be expressed in the form
AX = XΛ (9)
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues of A; that is
λ 1 0 · · · 0 0 λ 2 · · · 0 · · · · · · · · · · · · 0 0 · · · λn
It has been proved that the eigenvectors associated with distinct eigenvalues are linearly independent (Theorem 1. Hence, the matrix X will be nonsingular if the λi's are distinct. If both sides of equation 9 are multiplied by X− i 1 , the result is
X−^1 AX = Λ (11)
Thus, by use of a matrix of eigenvectors and its inverse, it is possible to transform any matrix A with distinct eigenvalues to a diagonal matrix whose diagonal elements are the eigenvalues of A. The transformation expressed by 11 is referred to as the diagonalization of matrix A. If the eigenvalues are not distinct, the diagonalization of matrix A may not be possible. For example, the matrix
A =
cannot be diagonalized as in 11. A matrix such as matrix A in equation 11 sometimes is spoken of as being similar to the diagonal matrix. In general, if there exists a nonsingular matrix C such that C−^1 AC = B for any two square matrices A and B of the same order, the A and B are called similar matrices, and the transformation of A to B is called a similarity transformation. Furthermore, if B is a diagonal matrix whose diagonal elements are the eigenvalues of A, the B is called the classical canonical form of matrix A. It is a unique matrix except for the order in which the eigenvalues appear along the principal diagonal.
If both sides of equation 16 are postmultiplied by X−^1 , the result is f (A) = 0, and the theorem is proved. Proofs of the Hamilton-Cayley Theorem for the general case without restric- tions on the eigenvalues of A may be found in most advanced texts on linear algebra. The Hamilton-Cayley Theorem may be applied to the problem of determining the inverse of a nonsingular matrix A. Let
c 0 λn^ + c 1 λn−^1 + · · · + cn− 1 λ + cn = 0 be the characteristic equation of A. Note that since A is a nonsingular matrix, λi 6 = 0; that is, every eigenvalue is nonzero, and cn 6 = 0. By the Hamilton-Cayley Theorem,
c 0 An^ + c 1 An−^1 + · · · + cn− 1 A + cnI = 0 and
I = −
cn
(c 0 An^ + c 1 An−^1 + · · · + cn− 1 A) (17)
If both sides of 18 are multiplied by A−^1 , the result is
A−^1 = −
cn
(c 0 An−^1 + c 1 An−^2 + · · · + cn− 1 I) (18)
Note that the calculation of an inverse by use of equation 18 is quite adaptable to high-speed digital computers and is not di cult to compute manually for small values of n. In calculating the powers of matrix A necessary in equation 18, the necessary information concerning tr(Ak) for calculating the c's is also obtained.
A−k, where k is positive integer, is de ned to be equal to
A−^1
)k
. By use of equation 18, it is now possible to express any negative integral power of a nonsingular matrix A of order n in terms of a linear function of the rst (n − 1) powers of A.
References
- Anthony J. Pettofrezzo. Matrices and Transformations. Dover Publications, June
