arXiv:2004.09826v1 [math.FA] 21 Apr 2020 rpsto 1.1. Proposition eko httomatrices two that know We bemti uhthat such T ible h qaematrix square The ito pca ust of subsets special of list r rm[ from are function matrix the paper, this nouoyif involutory λ T otof root fteeeit sequence a exists there if k 1 0 = λ , . . . , Let e od n phrases. and words Key 2010 = U , M ahmtc ujc Classification. Subject 1 arcscnhv elro rb nouoy vnuly ewl rep will form. we block Eventually, orth a involutory. in real be matrices that or idempotent demonstrate root real also have will can We matrices entries. real with matrices Abstract. S O I U P UTE RPRISO NOUOYAND INVOLUTORY OF PROPERTIES FURTHER A U ∈ , n n n n n 2 n n if , h e falivltr matrices, involutory all of set the : 5 h e fsmercmatrices, symmetric of set the : h ru faluiaymatrices, unitary all of group the : h e falieptn matrices. idempotent all of set the : . . . , h ru fotooa matrices, orthogonal of group The : uhthat such ,[ ], eoethe denote n , A A 9 A 2 ]. [ 1 = = ,[ ], = nti ae,w ildrv h elroso eti esof sets certain of roots real the derive will we paper, this In 1. C I. Let 4 U A 2 .Mti ucin aebe tde n[ in studied been have functions Matrix ]. A .MRAORADA MIRZAPOUR A. AND MIRZAPOUR F. nrdcinadPreliminaries and Introduction n tcnas esi that said be also can it and , ∗ nti ril,w edtefloigpooiin which propositions following the need we article, this In DMOETMATRICES IDEMPOTENT ssi ob dmoeto rjcin if projection, or idempotent be to said is diag( C = A ot nouay idempotent. involutary, root, ∗ M A − T ean be { − ler fall of algebra n λ C = : 1 1 diag( k A λ , . . . , } T n f and − uhthat such − ( 1 A λ BT qaecmlxmti.Then matrix. complex square 1 = ) n λ , . . . , B ) rmr 52;Scnay65F30. Secondary 15A24; Primary n ti ignlzbei hr exist there if diagonalizable is it and , U 1 √ rmteohrsd,matrix side, other the From . r iia,i hr xssa invert- an exists there if similar, are n A − n C o pcfi arcsi studied. is matrices specific for qaemtie.Floigi the is Following matrices. square ) k T −→ n ntrl ignlzbeif diagonalizable unitarily and I A and sro approximable - root is C k 2 k = 2 A ,[ ], ogonal resent A 2 3 o each for , = ,[ ], A, C 6 .In ]. and sa is 2 F.MIRZAPOURANDA.MIRZAPOUR

1. A is idempotent if and only if A is similar to a of the form diag(1, ..., 1, 0, ..., 0). 2. A is involutory if and only if A is similar to a diagonal matrix of the form diag(1, ..., 1, 1, ..., 1). − − Proposition 1.2. Let A and B be real square matrices of the same size. If P is a complex such that P −1AP = B, then there exists a real invertible matrix Q such that Q−1AQ = B.

Proposition 1.3. Let A and B be real square matrices of the same size. If A = UBU ∗ for some U, then there exists a real Q such that A = QBQT .

Proposition 1.4. Every real orthogonal matrix is real orthogonally similar to a direct sum of real orthogonal matrices of order at most 2.

2. matrices

In this section, we obtain some properties of the involutory matrices and by applying them we derive the real root of some special matrices. We start with matrices of order 2.

Lemma 2.1. The class of all real involutory matrices of order 2 is as follows: a b R 1 0 R 1−a2 ; a, b , b =0 ± ; c I2 . ( b a ! ∈ 6 ) ( c 1 ! ∈ ) {± } − [ ∓ [ 2 Lemma 2.2. The class of all real matrices like A such that A = I2 is as − follows: a b 2 − ; a, b R, b =0 . 1+a a ∈ 6 ( b − ! ) a b Herein after we refer to 2 − as Ψ(a, b). 1+a a b − ! Remark 2.3. In the lemma (2.1), if a 1 and b = √1 a2, then | | ≤ − cos θ sin θ ; θ R. sin θ cos θ ∈ − ! 1 0 1 0 Remark 2.4. ± are the only roots of . 0 i 0 1 ± ! − ! FURTHER PROPERTIES OF INVOLUTORY AND IDEMPOTENT MATRICES 3

Theorem 2.5. Suppose A is a real involutory matrix of order n and det A> 0, then A has a real root.

Proof. Since A is a real involutory matrix, then by propositions (1.1) and (1.2), there is an invertible real matrix B such that

− A = B 1diag(1,..., 1, 1,..., 1)B, − − thus det A = 1 or det A = 1. If det A = 1, then the number of eigenvalues − 1 is even. Therefore − C = diag(1,..., 1, 1,..., 1) − − 1 0 1 0 = Ik − − , ⊕ 0 1 ⊕···⊕ 0 1 − ! − ! has many real roots, for instance, for arbitrary real numbers a1,...,at and non-zero real numbers b1,...,bt, if

D = diag( 1, , 1) Ψ(a1, b1) Ψ(at, bt), ± ··· ± ⊕ ⊕···⊕ then A = B−1D2B =(B−1DB)2. 

Theorem 2.6. Let A be a real of size n with the eigenvalues

λ1 λn. If every negative eigenvalue is repeated twice or even times, ≥ ···≥ then A has a real root.

Proof. Given A is real symmetric, then by proposition (1.3) there is a real T orthogonal matrix Q such that A = Q diag(λ1,...,λn)Q. If λn 0, we have ≥ nothing to prove. Now if

λ1 λk 0 >λk+1 λn, ≥···≥ ≥ ≥···≥ then by assumption, each λj, k +1 j n is repeated twice or even times, ≤ ≤ therefore

λk+1 0 λn 0 C = diag(λ1, ,λk) . ··· ⊕ 0 λk+1 ! ⊕···⊕ 0 λn !

Thus for all real numbers ak+1,...,an and non-zero real numbers bk+1,...,bn,

D = diag( λ1, , λk) λk+1Ψ(ak+1, bk+1) λnΨ(an, bn), ± ··· ± ⊕ − ⊕···⊕ − are real rootsp of C and Ap=(QT pDQ)2. p  4 F.MIRZAPOURANDA.MIRZAPOUR

In this case, note that this matrix has no symmetrical root. According to proposition (1.4), for any real orthogonal matrix A there exist

α1,...,αs and β1 ...,βt such that A is similar to matrix

cos α1 sin α1 cos αs sin αs C = Ik Il ⊕ − ⊕ sin α1 cos α1 ⊕···⊕ sin αs cos αs − ! − ! cos β1 sin β1 cos β sin β t t . ⊕ sin β1 cos β1 ⊕···⊕ sin βt cos βt − ! − ! We have the following theorem for real orthogonal matrices.

Theorem 2.7. Suppose A is a real orthogonal matrix that is similar to the following matrix

cos α1 sin α1 cos αs sin αs C = Ik Il , ⊕ − ⊕ sin α1 cos α1 ⊕···⊕ sin αs cos αs − ! − ! where the eigenvalue 1 is repeated even times, then A has a real orthogonal − root.

Proof. By the assumption, there is a real orthogonal matrix P such that

T cos α1 sin α1 cos αs sin αs C = P AP = Ik Il , ⊕ − ⊕ sin α1 cos α1 ⊕···⊕ sin αs cos αs − ! − ! cos π sin π cos π sin π = Ik ⊕ sin π cos π ⊕···⊕ sin π cos π − ! − ! cos α1 sin α1 cos α sin α s s , ⊕ sin α1 cos α1 ⊕···⊕ sin αs cos αs − ! − ! if cos π sin π cos π sin π D = diag( 1, , 1) 2 2 2 2 ± ··· ± ⊕ sin π cos π ⊕···⊕ sin π cos π − 2 2 ! − 2 2 ! cos α1 sin α1 cos αs sin αs 2 2 2 2 , ⊕ sin α1 cos α1 ⊕···⊕ sin αs cos αs − 2 2 ! − 2 2 ! then A = PCP T = PD2P T =(PDP T )2, where PDP T is real orthogonal root of A.  FURTHER PROPERTIES OF INVOLUTORY AND IDEMPOTENT MATRICES 5

Remark 2.8. If π π π π cos 2k sin 2k cos 2k sin 2k Dk = Ir π π π π ⊕ sin k cos k ⊕···⊕ sin k cos k − 2 2 ! − 2 2 ! α1 α1 αs αs cos k sin k cos k sin k 2 2 2 2 , α1 α1 αs αs ⊕ sin k cos k ⊕···⊕ sin k cos k − 2 2 ! − 2 2 ! T 2k then Dk I and (PDkP ) = A, i.e. A is root-approximable. −→ Remark 2.9. With the above given if

T cos β1 sin β1 cos βt sin βt P AP = Ir Il , ⊕ − ⊕ sin β1 cos β1 ⊕···⊕ sin βt cos βt − ! − ! then A is an involutory matrix.

3. Idempotent matrices

By proposition (1.1), if P is an , then it is similar to IO where I is identity, i.e. there are matrices A,B,C and D such that OO ! A B A and D are square and A and I are of the same size, then M = CD ! is invertible and IO P = M M −1. OO ! XY If M −1 = and A is invertible, then we have U V ! − − − − − − − − X = A 1 + A 1B(D CA 1B) 1CA 1,Y = A 1B(D CA 1B) 1, − − − U = (D CA−1B)−1CA−1, V =(D CA−1B)−1, − − −

A B IO XY AX AY P = = CD ! OO ! U V ! CXCY ! I + B(D CA−1B)−1CA−1 B(D CA−1B)−1 = − − − , CA−1 + CA−1B(D CA−1B)−1CA−1 CA−1B(D CA−1B)−1 − − − ! and if D is invertible, we have

A−1 + A−1B(D CA−1B)−1CA−1 =(A BD−1C)−1. − − 6 F.MIRZAPOURANDA.MIRZAPOUR

Consider S =(D CA−1B)−1 and T =(A BD−1C)−1. We get − − AT BS P = − CT CA−1BS − ! AO TO O B OO = −1 . CO ! OO ! − O CA B ! O S ! Thus we have proved the following theorem:

Theorem 3.1. Let A and D be two invertible matrices of orders n and m respectively, while B and C are two matrices of orders n m and m n × × respectively. Then A(A BD−1C)−1 B(D CA−1B)−1 P = − − − C(A BD−1C)−1 CA−1B(D CA−1B)−1 − − − ! is an idempotent.

Example 3.2. Let a, b, c and d be real numbers, whereas bc = ad = 0, and 6 6 Im A = aIn, B = b ,C = c Im O ,D = dIm, (n m), O ! ≥  

Im aIn b M =  O !  ,  c Im O dIm        d a ad−bc Im O S = Im, T = 1 , ad bc O I − − a n m ! therefore

adIm O abIm 1 − P =  O (ad bc)In−m ! O !  ad bc − −  cdIm O bcIm   −  is idempotent and    

ad+bc 2 Im O abIm 2 ad−bc − T =2P I =  O 2 In−m ! O !  − ad bc − ad+bc  cdIm O 2 Im   −  is an involutory matrix.     FURTHER PROPERTIES OF INVOLUTORY AND IDEMPOTENT MATRICES 7

References

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Department of Mathematics, Faculty of Sciences, University of Zanjan, P.O. Box 45371-38791, Zanjan, Iran. E-mail address: [email protected]

Department of Electrical and Computer, Zanjan branch, Islamic Azad Uni- versity, Zanjan, Iran E-mail address: [email protected]