A square matrix is nondegenerate if and only if its rows columns are linearly independent.. There exist matrices A kwhose positive integer power is equal to the zero matrix, even if A≠O.
Trang 1Remark. If a matrix is real (i.e., all its entries are real), then the corresponding transpose and the adjoint
matrix coincide.
A square matrix A is said to be normal if A ∗ A = AA ∗ A normal matrix A is said to be
unitary if A ∗ A = AA ∗ = I, i.e., A ∗ = A– 1(see Paragraph 5.2.1-6).
5.2.1-4 Trace of a matrix
The trace of a square matrix A≡[a ij ] of size n×n is the sum S of its diagonal entries,
S = Tr(A) =
n
i=1
a ii.
If λ is a scalar and square matrices A and B has the same size, then
Tr(A + B) = Tr(A) + Tr(B), Tr(λA) = λTr(A), Tr(AB) = Tr(BA),
5.2.1-5 Linear dependence of row vectors (column vectors)
A row vector (column vector) B is a linear combination of row vectors (column vectors)
A1, , A k if there exist scalars α1, , α ksuch that
B = α1A1+· · · + α k A k.
Row vectors (column vectors) A1, , A k are said to be linearly dependent if there exist scalars α1, , α k (α21+· · · + α2
k≠ 0) such that
α1A1+· · · + α k A k = O,
where O is the zero row vector (column vector).
Row vectors (column vectors) A1, , A k are said to be linearly independent if, for any α1, , α k (α21+· · · + α2k≠ 0) we have
α1A1+· · · + α k A k≠O
THEOREM Row vectors (column vectors) A1, , A k are linearly dependent if and only if one of them is a linear combination of the others
5.2.1-6 Inverse matrices
Let A be a square matrix of size n×n , and let I be the unit matrix of the same size.
A square matrix B of size n×n is called a right inverse of A if AB = I A square matrix C of size n×n is called a left inverse of A if CA = I If one of the matrices B
or C exists, then the other exists, too, and these two matrices coincide In such a case, the matrix A is said to be nondegenerate (nonsingular).
THEOREM A square matrix is nondegenerate if and only if its rows (columns) are linearly independent
Remark. Generally, instead of the terms “left inverse matrix” and “right inverse matrix”, the term “inverse
matrix” is used with regard to the matrix B = A–1for a nondegenerate matrix A, since AB = BA = I.
UNIQUENESSTHEOREM The matrix A–1 is the unique matrix satisfying the condition
AA–1 = A– 1A = I for a given nondegenerate matrix A.
Remark For the existence theorem, see Paragraph 5.2.2-7.
Trang 2Properties of inverse matrices:
(AB)–1= B–1A–1, (λA)– 1= λ– 1A– 1,
(A–1)–1= A, (A–1)T = (A T)–1, (A–1)∗ = (A ∗)–1,
where square matrices A and B are assumed to be nondegenerate and scalar λ≠ 0
The problem of finding the inverse matrix is considered in Paragraphs 5.2.2-7, 5.2.4-5, and 5.5.2-3
5.2.1-7 Powers of matrices
A product of several matrices equal to one and the same matrix A can be written as a positive
integer power of the matrix A: AA = A2, AAA = A2A = A3, etc For a positive integer k, one defines A k = A k–1A as the kth power of A For a nondegenerate matrix A, one defines
A0= AA– 1= I, A–k = (A– 1)k Powers of a matrix have the following properties:
A p A q = A p+q, (A p)q = A pq,
where p and q are arbitrary positive integers and A is an arbitrary square matrix; or p and q are arbitrary integers and A is an arbitrary nondegenerate matrix.
There exist matrices A kwhose positive integer power is equal to the zero matrix, even
if A≠O If A k = O for some integer k >1, then A is called a nilpotent matrix
A matrix A is said to be involutive if it coincides with its inverse: A = A– 1or A2= I.
5.2.1-8 Polynomials and matrices Basic functions of matrices
A polynomial with matrix argument is the expression obtained from a scalar polynomial f (x)
by replacing the scalar argument x with a square matrix X:
f (X) = a0I + a1X + a2X2+· · · ,
where a i (i =0,1, 2, ) are real or complex coefficients The polynomial f (X) is a square
matrix of the same size as X.
A polynomial with matrix coefficients is an expression obtained from a polynomial f (x)
by replacing its coefficients a i (i =0, 1, 2, ) with matrices Ai (i =0, 1,2, ) of the same size:
F (x) = A0+ A1x + A2x2+· · ·
Example 3 For the matrix
A=
(4 –8 1
5 – 9 1
4 – 6 – 1
) ,
the characteristic matrix (see Paragraph 5.2.3-2) is a polynomial with matrix coefficients and argument λ:
F (λ)≡A – λI = A0+ A1λ=
(4– λ –8 1
5 – 9– λ 1
4 – 6 – 1– λ
) , where
A0= A =
(4 –8 1
5 – 9 1
4 – 6 – 1
) , A1= –I =
(–1 0 0
0 – 1 0
0 0 – 1
) The corresponding adjugate matrix (see Paragraph 5.2.2-7) can also be represented as a polynomial with matrix coefficients:
G (λ) =
(λ2 + 10λ+ 15 – 8λ– 14 λ+ 1
5λ+ 9 λ2– 3λ– 8 λ+ 1
4λ+ 6 – 6λ– 8 λ2+ 5λ+ 4
)
= A0+ A1λ + A2λ2,
Trang 3A0=
(15 –14 1
9 – 8 1
6 – 8 4
) , A1=
(10 –8 1
5 – 3 1
4 – 6 5
) , A2= I =
(1 0 0
0 1 0
0 0 1
)
The variable x in a polynomial with matrix coefficients can be replaced by a matrix X,
which yields a polynomial of matrix argument with matrix coefficients In this situation, one distinguishes between the “left” and the “right” values:
F (X) = A0+ A1X + A2X2+· · · ,
$
F (X) = A0+ XA1+ X2A2+· · ·
The exponential function of a square matrix X can be represented as the following
convergent series:
e X =1+ X + X2
2! +
X3 3! +· · · =
∞
k=0
k! . The inverse matrix has the form
(e X)–1= e–X =1– X + X
2 2! –
X3 3! +· · · =
∞
k=0
(–1)k X k
k! .
Remark. Note that e X e Y ≠e Y e X , in general The relation e X e Y = e X+Y holds only for commuting
matrices X and Y
Some other functions of matrices can be expressed in terms of the exponential function:
sin X = 1
2i (e iX – e–iX), cos X = 1
2(e iX + e–iX),
sinh X = 1
2(e X – e–X), cosh X =
1
2(e X + e–X).
5.2.1-9 Decomposition of matrices
THEOREM1 For any square matrix A, the matrix S1 = 12(A + A T) is symmetric and
the matrix S2 = 12(A – A T) is skew-symmetric The representation of A as the sum of symmetric and skew-symmetric matrices is unique: A = S1+ S2
THEOREM2 For any square matrix A, the matrices H1=12(A+A ∗)and H2= 21i (A–A ∗)
are Hermitian, and the matrix iH2is skew-Hermitian The representation of A as the sum
of Hermitian and skew-Hermitian matrices is unique: A = H1+ iH2
THEOREM3 For any square matrix A, the matrices AA ∗ and A ∗ A are nonnegative Hermitian matrices (see Paragraph 5.7.3-1)
THEOREM4 Any square matrix A admits a polar decomposition
A = QU and A = U1Q1,
where Q and Q1 are nonnegative Hermitian matrices, Q2 = AA ∗ and Q21 = A ∗ A , and U and U1 are unitary matrices The matrices Q and Q1 are always unique, while the matrices U and U1are unique only in the case of a nondegenerate A.
Trang 45.2.1-10 Block matrices.
Let us split a given matrix A≡[a ij ] (i =1, 2, , m; j =1, 2, , n) of size m×ninto
separate rectangular cells with the help of (M –1) horizontal and (N –1) vertical lines Each
cell is a matrix A αβ≡[a ij ] (i = i α , i α+1, , iα + m α–1; j = jβ , j β+1, , jβ + n β–1) of
size m α×n β and is called a block of the matrix A Here i α = m α–1+ i α–1, j β = n β–1+ j β–1
Then the given matrix A can be regarded as a new matrix whose entries are the blocks:
A≡[A αβ ] (α =1, 2, , M ; β =1, 2, , N ) This matrix is called a block matrix
Example 4 The matrix
A≡
⎛
⎜
⎜
a11 a12 a13 a14 a15
a21 a22 a23 a24 a25
a31 a32 a33 a34 a35
a41 a42 a43 a44 a45
a51 a52 a53 a54 a55
⎞
⎟
⎟
can be regarded as the block matrix
A≡A11 A12
A21 A22
of size 2 × 2 with the entries being the blocks
A11 ≡a11 a12 a13
a21 a22 a23
, A12 ≡a14 a15
a24 a25
,
A21 ≡
(
a31 a32 a33
a41 a42 a43
a51 a52 a53
) , A22 ≡
(
a34 a35
a44 a45
a54 a55
)
of size 2 × 3 , 2 × 2 , 3 × 3 , 3 × 2 , respectively.
Basic operations with block matrices are practically the same as those with common matrices, the role of the entries being played by blocks:
1 For matrices A ≡[a ij] ≡[A αβ ] and B ≡[b ij] ≡[B αβ] of the same size and the same
block structure, their sum C ≡[C αβ ] = [A αβ + B αβ] is a matrix of the same size and the same block structure
2 For a matrix A≡[a ij ] of size m× n regarded as a block matrix A≡[A αβ ] of size M×N,
the multiplication by a scalar is defined by λA = [λA αβ ] = [λa ij]
3 Let A ≡ [a ik] ≡ [A αγ ] and B ≡ [b kj] ≡ [B γβ] be two block matrices such that the
number of columns of each block A αγ is equal to the number of the rows of the
block B γβ Then the product of the matrices A and B can be regarded as the block matrix C≡[C αβ] = [
γ A αγ B γβ].
4 For a matrix A≡[a ij ] of size m× n regarded as a block matrix A≡[A αβ ] of size M×N,
the transpose has the form A T = [A T βα]
5 For a matrix A≡[a ij ] of size m× n regarded as a block matrix A≡[A αβ ] of size M×N,
the adjoint matrix has the form A ∗ = [A ∗ βα]
Let A be a nondegenerate matrix of size n×nrepresented as the block matrix
,
where A11 and A22 are square matrices of size p×p and q×q , respectively (p + q = n) Then the following relations, called the Frobenius formulas, hold:
A–1=
A–1
11+ A–111A12N A21A–111 –A–111A12N
–N A21A–1
,
A–1=
22
–A–221A21K A–1
22+ A–221A21KA12A–221
Here, N = (A22– A21A–1
11A12)–1, K = (A11– A12A–1
22A21)–1; in the first formula, the matrix
A11is assumed nondegenerate, and in the second formula, A22is assumed nondegenerate.
Trang 5The direct sum of two square matrices A and B of size m×m and n×n, respectively,
is the block matrix C = A⊕B=
A 0
0 B
of size m + n.
Properties of the direct sum of matrices:
1 For any square matrices A, B, and C the following relations hold:
(A⊕B)⊕C = A⊕(B⊕C) (associativity),
Tr(A⊕B ) = Tr(A) + Tr(B) (trace property)
2 For nondegenerate square matrices A and B, the following relation holds:
(A⊕B)–1= A–1⊕B–1.
3 For square matrices A m , B m of size m×m and square matrices A n , B n of size n×n, the following relations hold:
(A m⊕A n ) + (B m⊕B n ) = (A m + B m)⊕(A n + B n);
(A m⊕A n )(B m⊕B n ) = A m B m⊕A n B n.
5.2.1-11 Kronecker product of matrices
The Kronecker product of two matrices A≡ [a iaja ] and B ≡ [b ibjb ] of size m a×n a and
m b×n b , respectively, is the matrix C ≡[c kh ] of size m a m b×n a n b with entries
c kh = a iaja b ibjb (k =1, 2, , ma m b ; h =1, 2, , na n b),
where the index k is the serial number of a pair (i a , i b) in the sequence (1,1), (1,2), , (1, mb), (2,1), (2,2), (ma , m b ), and the index h is the serial number of a pair (j a , j b
in a similar sequence This Kronecker product can be represented as the block matrix
C≡[a iaja B]
Note that if A and B are square matrices and the number of rows in C is equal to the number of rows in A, and the number of rows in D is equal to the number of rows in B, then
(A⊗B )(C⊗D ) = AC⊗BD The following relations hold:
(A⊗B)T = A T ⊗B T, Tr(A⊗B ) = Tr(A)Tr(B).
5.2.2 Determinants
5.2.2-1 Notion of determinant
With each square matrix A≡[a ij ] of size n×none can associate a numerical characteristic,
called its determinant The determinant of such a matrix can be defined by induction with respect to the size n.
For a matrix of size1 × 1(n =1), the first-order determinant is equal to its only entry,
Δ≡det A = a11 For a matrix of size 2 × 2 (n = 2), the second-order determinant, is equal to the difference of the product of its entries on the main diagonal and the product of its entries
on the secondary diagonal:
Δ≡det A≡a11 a12
a21 a22
= a11a22– a12a21.
Trang 6For a matrix of size3 × 3(n =3), the third-order determinant,
Δ≡det A≡
= a11a22a33+ a12a23a31+ a21a32a13– a13a22a31– a12a21a33– a23a32a11.
This expression is obtained by the triangle rule (Sarrus scheme), illustrated by the following
diagrams, where entries occurring in the same product with a given sign are joined by segments:
+
@
@
@
@
@
@
@
@
–
HHH
H
A A A
AA H H H H A A A
For a matrix of size n×n (n >2), the nth-order determinant is defined as follows under
the assumption that the (n –1)st-order determinant has already been defined for a matrix of
size (n –1)×(n –1)
Consider a matrix A = [a ij ] of size n×n The minor M j i corresponding to an entry a ij
is defined as the (n –1)st-order determinant of the matrix of size (n –1)×(n –1) obtained
from the original matrix A by removing the ith row and the jth column (i.e., the row and the column whose intersection contains the entry a ij ) The cofactor A i j of the entry a ij is
defined by A i j = (–1)i+j M i
j (i.e., it coincides with the corresponding minor if i + j is even,
and is the opposite of the minor if i + j is odd).
The nth-order determinant of the matrix A is defined by
Δ≡det A≡
a11 a12 · · · a1n
a21 a22 · · · a2n
. .
a n1 a n2 · · · a nn
=
n
k=1
a ik A i
k =
n
k=1
a kj A k
j.
This formula is also called the ith row expansion of the determinant of A and also the jth
column expansion of the determinant of A.
Example 1 Let us find the third-order determinant of the matrix
A=
(1 –1 2
6 1 5
2 – 1 – 4
)
To this end, we use the second-column expansion of the determinant:
det A =
3
i=1
(– 1 )i+2a i M i= (– 1 )1+2× (– 1 ) × 6 5
2 – 4 + (–1)2 + 2 × 1 × 1 2
2 – 4 + (–1)3 + 2 × (– 1 ) × 1 2
6 5
= 1 × [ 6 × (– 4 ) – 5 × 2 ] + 1 × [ 1 × (– 4 ) – 2 × 2 ] + 1 × [ 1 × 5 – 2 × 6 ] = – 49
5.2.2-2 Properties of determinants
Basic properties:
1 Invariance with respect to transposition of matrices:
det A = det A T
2 Antisymmetry with respect to the permutation of two rows (or columns): if two rows (columns) of a matrix are interchanged, its determinant preserves its absolute value, but changes its sign
Trang 73 Linearity with respect to a row (or column) of the corresponding matrix: suppose
that the ith row of a matrix A ≡ [a ij] is a linear combination of two row vectors,
(a i1, , a i3) = λ(b1, , b n ) + μ(c1, , c n); then
det A = λ det A b + μ det A c,
where A b and A c are the matrices obtained from A by replacing its ith row with (b1, , b n ) and (c1, , c n) This fact, together with the first property, implies that a
similar linearity relation holds if a column of the matrix A is a linear combination of
two column vectors
Some useful corollaries from the basic properties:
1 The determinant of a matrix with two equal rows (columns) is equal to zero
2 If all entries of a row are multiplied by λ, the determinant of the resulting matrix is multiplied by λ.
3 If a matrix contains a row (columns) consisting of zeroes, then its determinant is equal
to zero
4 If a matrix has two proportional rows (columns), its determinant is equal to zero
5 If a matrix has a row (column) that is a linear combination of its other rows (columns), its determinant is equal to zero
6 The determinant of a matrix does not change if a linear combination of some of its rows
is added to another row of that matrix
THEOREM(NECESSARY AND SUFFICIENT CONDITION FOR A MATRIX TO BE DEGENER
-ATE) The determinant of a square matrix is equal to zero if and only if its rows (columns) are linearly dependent
5.2.2-3 Minors Basic minors Rank and defect of a matrix
Let A ≡[a ij ] be a matrix of size n×n Its mth-order (m ≤ n ) minor of the first kind,
denoted by M i1i2 im
j1j2 jm , is the mth-order determinant of a submatrix obtained from A by
removing some of its n – m rows and n – m columns Here, i1, i2, , i m are the
indices of the rows and j1, j2, , j m are the indices of the columns involved in that
submatrix The (n – m)th-order determinant of the second kind, denoted by M i j11i j22 im jm, is
the (n – m)th-order determinant of the submatrix obtained from A by removing the rows and the columns involved in M i1i2 im
j1j2 jm The cofactor of the minor M j i11j i22 im jm is defined by
A i1i2 im
j1j2 jm= (–1)i1 +i 2 +···+im+j 1 +j 2 +···+jmM i1i2 im
j1j2 jm.
Remark. minors of the first kind can be introduced for any rectangular matrix A≡[a ij ] of size m×n Its
k th-order (k≤ min {m , n}) minor M i1i2 i k
j1j2 j k is the determinant of the submatrix obtained from A by removing some of its m – k rows and n – k columns.
LAPLACE THEOREM Given m rows with indices i1, , i m (or m columns with indices
j1, , j m ) of a square matrix A, its determinantΔ is equal to the sum of products of all
m th-order minors M i1i2 im
j1j2 jm in those rows (resp., columns) and their cofactors A i1i2 im
j1j2 jm, i.e.,
Δ≡det A =
j1 ,j2 , ,jm
M i1i2 im
j1j2 jm A i j11i j22 im jm =
i1 ,i2 , ,im
M i1i2 im
j1j2 jm A i j11i j22 im jm.
Here, in the first sum i1, , i m are fixed, and in the second sum j1, , j mare fixed
Let A≡[a ij ] be a matrix of size m×nwith at least one nonzero entry Then there is a
positive integer r≤nfor which the following conditions hold:
i) the matrix A has an rth-order nonzero minor, and
ii) any minor of A of order (r +1) and higher (of it exists) is equal to zero