C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 37–43 (2017) D O I: 10.1501/C om mua1_ 0000000798 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
PSEUDO MATRIX MULTIPLICATION
OSMAN KEÇ·IL·IO ¼GLU AND HAL·IT GÜNDO ¼GAN
Abstract. In this paper, a new matrix multiplication is de…ned in Rm;n Rn;p by using scalar product in Rn, where Rm;nis set of matrices of m rows and n columns. With this multiplication it has been shown that Rn;n is an algebra with unit. By considering this new multiplication we de…ne eigenvalues and eigenvectors of square n nmatrix A and also present some applications.
1. Introduction
In [1], Lorentzian matrix multiplication was introduced. For some applications related to this multiplication, we refer the papers [2-5].
In the present paper we aim to de…ne a new matrix multiplication using scalar product on Rn of which index is : We generalize some properties given for
ordi-nary matrix multiplication. As one of the most important properties of this new multiplication we don’t need to use sign matrix to obtain orthogonal, symmetric matrix etc. In the third section, we examine the concepts of eigenvalues and its eigenvectors of square n n matrix A. Finally, we study on diagonalizable matrix.
We start with some basic concepts and notations.
Let Rm;nbe the set of all m n matrices. Rm;nwith the matrix addition and the scalar-matrix multiplication is a real vector space. More properties of the ordinary matrix multiplication can be found in [7].
Let Rn be pseudo-Euclidean space over the real …eld R equipped with a scalar
product hx; yi which is symmetric, non degenerate bilinear form;
hx; yi = X i=1 xiyi+ n X i= +1 xiyi
where x; y 2 Rn and is an integer with 0 n [8].
Received by the editors: July 28, 2016; Accepted: November 04, 2016. 2010 Mathematics Subject Classi…cation. 15A18.
Key words and phrases. p-matrix multiplication, eigenvalue, eigenvector.
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2. Pseudo Matrix Multiplication and Properties
Let A1; : : : ; Am denote the row vectors of A = [aij] 2 Rm;n and B1; : : : ; Bp
denote the column vectors of B = [bjk] 2 Rn;p: Then we de…ne a new matrix
multiplication denoted by “ ”, as A B = 2 6 6 6 4 hA1; B1i hA1; B2i hA1; Bpi hA2; B1i hA2; B2i hA2; Bpi .. . ... ... hAm; B1i hAm; B2i hAm; Bpi 3 7 7 7 5 = 2 4 X j=1 aijbjk+ n X j= +1 aijbjk 3 5 :
We call this multiplication as pseudo matrix multiplication and if we let Ai to be
ith row of A and Bj to be jth column of B then (i; j) entry of A B is A i; Bj .
Note that A B is an m p matrix. We will denote Rm;n with pseudo matrix
multiplication by Rm;n. In the special case of we get followings:
(1) For = 0; A B coincide with usual matrix multiplication.
(2) For = 1; A B coincide with Lorentzian matrix multiplication de…ned in [1].
Also, in the results and de…nitions given in throughout the paper one can easily obtain the classical ones when = 0:
In the sequel we present some properties of new type matrix multiplication and give the analogous of de…nitions, in the classical matrix multiplication, by . Theorem 1. The following statements are satis…ed.
i) For every A 2 Rm;n; B 2 Rn;p; C 2 Rp;r; A (B C) = (A B) LC
ii) For every A 2 Rm;n; B; C 2 Rn;p; A (B + C) = A B + A C
iii) For every A; B 2 Rm;n; C 2 Rn;p; (A + B) C = A C + B C
iv) For every k 2 R; A 2 Rm;n; B 2 Rn;p; k(A B) = (kA) B = A (kB)
De…nition 1. n n identity matrix according to pseudo matrix multiplication, denoted by In= [{ij] ; is de…ned by {ij = 8 < : 1 ; i = j and 1 i; j 1 ; i = j and + 1 i; j n 0 ; i 6= j ;
that is In= 2 6 6 6 6 6 6 6 6 4 1 0 0 0 .. . . .. ... ... . .. ... 0 1 0 0 0 0 1 0 .. . . .. ... ... . .. ... 0 0 0 1 3 7 7 7 7 7 7 7 7 5 :
Note that for every A 2 Rm;n; I
m A = A In= A:
Corollary 1. Rn;n with pseudo matrix multiplication is an algebra with unit. De…nition 2. An n n matrix A is called p-invertible, if there exists an n n matrix B such that A B = B A = In. Then B is called p-inverse of A and is
shown by A 1.
De…nition 3. Let A = [aij] be an m n matrix. Then the transpose of A is the
n m matrix AT obtained by interchanging the rows and columns of A;so that the
(i; j) th entry of AT is a ji:
Theorem 2. Let A and B be matrices of the appropriate sizes so that the following operations make sense, and c be a scalar. Then
(1) (A + B)T = AT+ BT
(2) A BT = BT AT
(3) (cA)T = cAT (4) AT T = A:
De…nition 4. Let A 2 Rn;n: If AT = A; AT = A and A 1= AT then A is said
to be symmetric, skew-symmetric and p-orthogonal matrix, respectively. Based on this de…nition, we obtain the following result.
Theorem 3. Let A 2 Rn;n: Then
(1) A is p-orthogonal if and only if the row vectors of A form an orthonormal basis of Rn under the scalar product; and
(2) A is p-orthogonal if and only if the column vectors of A form an orthonor-mal basis of Rn under the scalar product.
Proof. We shall only prove (1), since the proof of (2) is almost identical. Let A1; : : : ; An denote the row vectors of A. Then
A AT = 2 6 4 hA1; A1i hA1; Ani .. . ... hAn; A1i hAn; Ani 3 7 5 :
It follows that A AT = I
n if and only if for every i; j = 1; : : : ; n
hAi; Aji = 8 < : 1 ; i = j and 1 i; j 1 ; i = j and + 1 i; j n 0 ; i 6= j : Then fA1; : : : ; Ang is an orthonormal basis of Rn:
De…nition 5. The determinant of a matrix A = [aij] 2 Rn;n is denoted by det A
and de…ned as
det A = X
2Sn
s( )a (1)1a (2)2 a (n)n;
where Sn is set of all permutations of the set f1; 2; ; ng and s( ) is sign of the
permutation .
Theorem 4. For every A; B 2 Rn;n; det(A B) = ( 1) det A det B:
Proof. Let A = [aij] ; B = [bjk] and A B = C. Let us denote the ith column of
matrix A by Ai and the kth column of matrix C by Ck
Ck= b1kA1 b2kA2 b kA + b( +1)kA +1+ + bnkAn; 1 k n: Then det(A B) = det[ b11A1 b21A2 b 1A + b( +1)1A +1+ + bn1An; : : : ; b1nA1 b2nA2 b nA + b( +1)nA +1+ + bnnAn] = X 2Sn ( 1) b (1)1b (2)2 b (n)ndet[A (1); A (2); ; A (n)] = ( 1) X 2Sn s ( ) b (1)1b (2)2 b (n)ndet[A1; A2; ; An] = ( 1) det A X 2Sn s ( ) b (1)1b (2)2 b (n)n = ( 1) det A det B:
3. Some Applications Of Pseudo Matrix Multiplication
Eigenvalues and eigenvectors play an important role in matrix theory because of its application in the areas of mathematics, physics and engineering. By this aim, we de…ne the eigenvalues and eigenvectors of square n n matrix A by pseudo matrix multiplication.
De…nition 6. Let A 2 Rn;n: An eigenvector of A is a nonzero vector x in Rn such
that
for some scalar : The scalar is called an eigenvalue of the matrix A, and we say that the vector x is an eigenvector belonging to the eigenvalue .
Theorem 5. The eigenvectors of a symmetric matrix A 2 Rn;n corresponding to
di¤ erent eigenvalues are orthogonal to each other.
Proof. For the eigenvectors x; y corresponding to two di¤erent eigenvalues ; of the matrix A, we can say that A x = x and A y = y; so
yT A x = yT x = hx; yi : (3.1)
But numbers are always their own transpose, so
yT A x = xT A y
= xT y
= xT y
yT A x = hx; yi : (3.2)
From (3:1) and (3:2), we get
( ) hx; yi = 0:
So = or hx; yi = 0, and it isn’t the former, so x and y are orthogonal. Example 1. Let A = 2 4 11 12 01 0 1 1 3 5 2 R3;3 2 :
A is a symmetric matrix. Then the eigenvalues of A are 1= 0; 2 =
p
2 1 and
3=
p
2 1: Some eigenvectors of A corresponding to 1; 2 and 3 are
u1= 2 4 11 1 3 5 ; u2= 2 4 1 + p 2 2 +p2 1 3 5 ; u3= 2 4 1 p 2 2 p2 1 3 5 respectively. For i 6= j; we get
hui; uji2= uTi 2uj = 0:
Then fu1; u2; u3g form an orthogonal basis of R32:
De…nition 7. A matrix A is diagonalizable if there exists a nonsingular matrix P and a diagonal matrix D such that
D = P 1 A P:
Theorem 6. Let all the eigenvalues of A 2 Rn;n are real. Then A diagonalizable
Proof. Let x1; : : : ; xn be n linearly independent eigenvectors of A associated with
the eigenvalues 1; : : : ; n: That is,
A xi= ixi ; i = 1; : : : ; n :
Now, we denote P = x1 xn : Since the columns of P are linearly
inde-pendent, P is invertible. Let D be diag [ 1; : : : ; ; +1; : : : ; n]. Then
A P = A x1 xn
= 1x1 nxn
= x1 xn diag [ 1; : : : ; ; +1; : : : ; n]
= P D:
Since A P = P D; it follows that D = P 1 A P which shows that A is
diagonalizable.
To prove the other direction we assume that A is diagonalizable. Then there ex-ists a nonsingular matrix P and a diagonal matrix D = diag [ 1; : : : ; ; +1; : : : ; n]
such that
D = P 1 A P:
If we multiply above equation with P from the left, we get
A P = P D (3.3)
which implies
A vi= ivi ; i = 1; : : : ; n (3.4)
where viare columns of P: The equations (3:4) show that v1; : : : ; vnare eigenvectors
of A corresponding to eigenvalues 1; : : : ; n: Furthermore, since P is invertible,
fv1; : : : ; vng are linearly independent.
Example 2. Consider the matrix A = 2 4 11 20 01 1 2 2 3 5 2 R3:3 2 :
The eigenvalues of A are 1= 2; 2= 0, 3= 1 and eigenvectors corresponding
these eigenvalues are u1= 2 4 21 0 3 5 ; u2= 2 4 21 2 3 5 ; u3= 2 4 10 1 3 5 respectively. Therefore P = 2 4 21 21 10 0 2 1 3 5 and P 1= 2 4 1 2 0 1 2 1 2 1 1 2 1 2 2 3 5 :
Finally P 1 A P = 2 4 20 00 00 0 0 1 3 5 : References
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Current address : Department of Statistics, K¬r¬kkale University, K¬r¬kkale, TURKEY E-mail address, Osman Keçilio¼glu: okecilioglu@yahoo.com
