矩阵的转置和一些特殊矩阵(英文)

T ' n n
'
'
bij b ji , so the (i, j)-entry of ( AB)T B T AT .
B T AT
is
bik akj bki a jk a jk bki , thus,
k 1 k 1 k 1
Although the transpose of the matrix can be defined in any number field, but for complex matrices, the more useful are the following so called conjugate transpose matrices. A matrix whose entries is obtained by taking their corresponding conjugate entries of A is called the conjugate matrix of A, written A . It is easy to prove,
Diagonal matrix
Scalar matrix Matrices
The diagonal matrix referred to just now is also written as diag( a1, a2, ...an ). taking the following forms are collectively called triangular matrix.
Definition3.3.3 A is a square matrix. (1)if A A , then call A symmetric matrix. T (2)if A - A , then call A skew-symmetric matrix. H (3)if A A , then call A Hermite matrix. H (4)if A - A ,then call A skew-Hermite matrix.
§3.3 Definition3.3.1
Transposition and some special matrices The transpose of A, written A , is the n m
T
Suppose that A= (a ij ) mn .
matrix whose rows are just the columns of A in the same order. Namely,
AT (a ji ) nm
a11 a 12 . . . a1n
a21 . . . am1 a22 . . . am 2 . . . . . . a2 n . . . amn
Proposition3.3.1 Let A and B denote matrices of the same size of n m , then (1) ( A ) A ;
T
Suppose that A= (a ij ) is an n n matrix, then we can know from definition, 1. A is a symmetric(Hermite) matrix aij a ji ( aij a ji ) . 2. A is a skew-symmetric(skew-Hermite)matrix aij -a ji ( aij -a ji ) . Therefore, symmetric and skew-symmetric matrices take the forms respectively:
n
a
k 1
n
ik kj
b , thus the (i, j)-entry of ( AB)T
c ji a jk bki .
k 1 '
（aij ） And then, suppose that A nm
T n '
'
（bij ） , B pn , then aij a ji ,
a11 0 . . . 0
a12 a22 .
. . . . .
. . . . .
.
a1n a2 n . . . . 0 ann . .
a11 a 21 . . . an1
0 a22 .
. . . .
A B A B , cA c A ,
AT A .
Definition3.3.2
H
T
The transpose of A’s conjugate matrix A
T
is called the conjugate transpose of
A, written A . Conjugate transpose matrices also have such conclusions as proposition 3.3.1. We do not add unnecessary statements any more. We introduce several common matrices of special types in the following.
T T
5. Suppose that A is an upper triangular matrix, and if A is also an normal matrix,then A is an diagonal matrix. 6. Show that the multiplication of the two orthogonal matrices is still an orthogonal matrix, and explain if the corresponding proposition is tenable in unitary matrix. 7. Show that the multiplication of two symmetric matrices are symmetric matrix if and only if they are commutative, and explain if the corresponding proposition is tenable in Hermite matrix. 8. Show that the trace of the multiplication of a symmetric matrix and a skew-symmetric matrix is 0. 9. Suppose that A is a nonzero real n 1 column matrix, A I an orthogonal matrix. 10. Suppose that A is an real matrix, if tr( AA )=0, then A=0. Illustrate this is not tenable to complex matrices; For complex matrices, what is the corresponding proposition? Show your conclusions. 11. Suppose that A is a diagonal matrix and entries in its diagonal are all different, then matrices that are commutative with A must be a diagonal matrix. 12. Suppose that A is an n n matrix, show that A is commutative with all n n matrices if and only if A is a scalar matrix. 13. Suppose that A is a square matrix, show that . (a).There are only one symmetric matrix B and only one skew-symmetric matrix such that A=B+C; (b).There are only one Hermite matrix B and only one skew-Hermite matrix C such that A=B+C; (c).There are only one couple of Hermite matrix B,C such that A=B+iC; 14. Suppose that A=B+iC, where B, C are all Hermite matrices, show that A is an normal matrix if and only if BC=CB. 15. Suppose that A, B are all n n real symmetric matrices, C is an n n real skew-symmetric matrix, and A B C , then A=B=C=0, show that if the corresponding
H
Easy to know
cos x sin x is just an orthogonal matrix. sin x cos x
H H
Matrices meet the requirement A A AA are called normal matrices. Obviously, unitary matrices, Hermite matrices and skew-Hermite matrices (particularly, orthogonal matrices, real symmetric matrices and real skew-symmetric matrices)are all normal matrices. Definition3.3.5 Suppose that A is a square matrix. (1)if there is a positive integer k such that A 0 , then call A nilpotent matrix.
an 2
. . . . . . .
. . . . .
0 0 . . . ann
Upper triangular matrix
Lower triangular matrix
Suppose that A= (a ij ) is an n n square matrix, then 1.A is a triangular matrix aij 0(i j ) 2.A is an upper triangular matrix aij 0(i j ) 3.A is a lower triangular matrix aij 0(i j ) Obviously, I n and Onn are all scalar matrices;
H
Scalar matrices are all diagonal matrices;
Diagonal matrices are not just triangular, but also symmetric matrices. Definition3.3.4 If A is a complex square, and A A AA I , then we call A unitary matrix. Unitary matrices whose entries are real number are called orthogonal matrices.
Symmetric matrix
Skew-symmetric matrix
Matrices whose entries are all zero except for those on the main diagonal are called diagonal matrices; Diagonal matrices whose entries on the main diagonal are the same are called scalar matrices. Scalar matrices whose entries on the diagonal are “a” are just aI.
k
(2)if A A , then call A idempotent matrix. 2 (3)if A I , then call A involutory matrix.
2
Exercise 1. Find out all the 2 2 orthogonal matrices. 2. For any matrix A, show that A A and AA exist and are all symmetric matrices. 3. Show that the multiplication of the two upper triangular matrices is still an upper triangular matrix. 4. Suppose that A is an upper triangular matrix, f(x) is a polynomial, show that f(A) is also an upper triangular matrix.;if entries of A in its main diagonal are successively a1, a2, ...an , then entries of f(A) are f ( a1 ), f ( a2 ),... f ( an ) .
a1 b . . . c
b a2 .
. . . . .
. . . . .
.
.
. . d
c . . . d an
Fra Baidu bibliotek
0 a - a 0 . . . . b .
. . . . .
. . . . .
b . . . . . c c 0 .
T T
(2) ( A B ) A B ;
T T T
(3) ( AB ) B A ;
T T T
Proof
since (1) and (2) are obvious, we need only prove property (3).
If A= (a ij ) mn , is
B= (b ij ) n p , so, AB= (cij ) m p , wherein cij
a1 0 . . . 0
0 a2 .
. . . . .
. . . . .
.
0 . . . . . 0 0 an .
a 0 0 a . . . . 0 .
. . . . .
. . . . .
0 . . . . . 0 0 a .