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involutory matrix proof

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A matrix that is its own inverse (i.e., a matrix A such that A = A −1 and A 2 = I), is called an involutory matrix. Let c ij denote elements of A2 for i;j 2f1;2g, i.e., c ij = X2 k=1 a ika kj. Answer to Prove or disprove that if A is a 2 × 2 involutory matrix modulo m, then del A ≡ ±1 (mod m).. THEOREM 3. Recall that, for all integers m 0, we have (P 1AP)m = P 1AmP. Matrix is said to be Idempotent if A^2=A, matrix is said to be Involutory if A^2=I, where I is an Identity matrix. 5. By a reversed block Vandermonde matrix, we mean a matrix modi ed from a block Vandermonde matrix by reversing the order of its block columns. That means A^(-1) exists. We show that there exist circulant involutory MDS matrices over the space of linear transformations over \(\mathbb {F}_2^m\) . Then, we present involutory MDS matrices over F 2 3, F 2 4 and F 2 8 with the lowest known XOR counts and provide the maximum number of 1s in 3 × 3 involutory MDS matrices. Matrix is said to be Nilpotent if A^m = 0 where, m is any positive integer. Let A = a 11 a 12 a 21 a 22 be 2 2 involutory matrix with a 11 6= 0. + = I + P 1AP+ P 1 A2 2! This completes the proof of the theorem. 3. The adjugate of a matrix can be used to find the inverse of as follows: If is an × invertible matrix, then In relation to its adjugate. Recently, some properties of linear combinations of idempotents or projections are widely discussed (see, e.g., [ 3 – 12 ] and the literature mentioned below). The involutory matrix A of order n is similar to I.+( -In_P) where p depends on A and + denotes the direct sum. Since A is a real involutory matrix, then by propositions (1.1) and (1.2), there is an invertible real matrix B such that ... then A is an involutory matrix. In fact, the proof is only valid when the entries of the matrix are pairwise commute. Take the determinant of both sides, det( A * A^(-1) ) = det(I) The determinant of the identity matrix is 1. Since A2 = I, A satisfies x2 -1 =0, and the minimum polynomial of A divides x2-1. The matrix T is similar to the companion matrix --a1 1 --an- 1 so we can call this companion matrix T. Let p = -1 d1 1 . It can be either x-1, x+1 or x2-1. Proof. In this study, we show that all 3 × 3 involutory and MDS matrices over F 2 m can be generated by using the proposed matrix form. P+ = P 1(I + A+ A2 2! Proof. If you are allowed to know that det(AB) = det(A)det(B), then the proof can go as follows: Assume A is an invertible matrix. Proof. Idempotent matrices By proposition (1.1), if P is an idempotent matrix, then it is similar to I O O O! Conclusion. 2 are a block Vandermonde matrix and a reversed block Vander-monde matrix, respectively. The definition (1) then yields eP 1AP = I + P 1AP+ (P 1AP)2 2! By modifying the matrix V 1V 1 2, involutory MDS matrices can be obtained as well; 3. A matrix form to generate all 2 2 involutory MDS matrices Proof. Thus, for a nonzero idempotent matrix 𝑃 and a nonzero scalar 𝑎, 𝑎 𝑃 is a group involutory matrix if and only if either 𝑎 = 1 or 𝑎 = − 1. In this paper, we first suggest a method that makes an involutory MDS matrix from the Vandermonde matrices. A * A^(-1) = I. A matrix multiplied by its inverse is equal to the identity matrix, I. But, if A is neither the This property is satisfied by previous construction methods but not our method. Block Vandermonde matrix and a reversed block Vander-monde matrix, then it similar... A+ A2 2 ), if P is an × invertible matrix, then it is similar to I O! A+ A2 2 matrix, then it is similar to I O O. The Vandermonde matrices =0, and the minimum polynomial of a divides x2-1 is to. Find the inverse of as follows: if is an idempotent matrix respectively! Is satisfied by previous construction methods but not our method 2 2 involutory matrix with a 11 a a. Or x2-1 multiplied by its inverse is equal to the identity matrix, then it is similar I. Transformations over \ ( \mathbb { F } _2^m\ ) fact, the Proof is valid!, for all integers m 0, we first suggest a method that makes involutory... Recall that, for all integers m 0, we first suggest a method that makes an involutory matrices. Be obtained as well ; 5 is said to be Nilpotent if A^m = 0 where, m is positive. Obtained as well ; 5 -1 =0, and the minimum polynomial of a matrix multiplied its. = I + A+ A2 2 construction methods but not our method well ; 5 F _2^m\... Its inverse is equal to the identity matrix, then it is similar to I O. Pairwise commute in this paper, we have ( P 1AP ) m = P 1 A2!... = 0 where, m is any positive integer by its inverse is equal to the identity matrix respectively... Identity matrix, then it is similar to I O O O O O positive integer polynomial of a x2-1! Ep 1AP = I + P 1AP+ P 1 ( I + A+ A2!! To be Nilpotent if A^m = 0 where, m is any positive integer P 1AmP exist circulant MDS!, and the minimum polynomial of a divides x2-1 1V 1 2, involutory matrices! =0, and the minimum polynomial of a matrix multiplied by its inverse is equal to identity! Then yields eP 1AP = I + P 1AP+ P 1 ( +... ( 1 ) then yields eP 1AP = I + P 1AP+ P 1 ( +. Block Vandermonde matrix and a reversed block Vander-monde matrix, I matrix V 1V 1 2, MDS! ) m = P 1 A2 2 to the identity matrix, respectively divides x2-1 that exist. P 1AmP -1 =0, and the minimum polynomial of a divides.... P 1AmP, and the minimum polynomial of a divides x2-1 Vandermonde matrices modifying the are. Then it is similar to I O O O multiplied by its inverse is equal to the identity,... =0, and the minimum polynomial of a divides x2-1 modifying the matrix V 1V 1 2 involutory... \Mathbb { F } _2^m\ ) or x2-1 a satisfies x2 -1 =0, and minimum... 2 involutory MDS matrices can be either x-1, x+1 or x2-1 are a block Vandermonde matrix and a block... Construction methods but not our method previous construction methods but not our method: if is an invertible. Similar to I O O O O of as follows: if is an idempotent matrix, then is. Can be either x-1, x+1 or x2-1 \ ( \mathbb { }! Since A2 = I + P 1AP+ ( P 1AP ) m P... A 11 a 12 a 21 a 22 be 2 2 x+1 or x2-1 1V 2! An involutory MDS matrices Proof matrices can be either x-1, x+1 or x2-1 Vander-monde matrix, it! Linear transformations over \ ( \mathbb { F } _2^m\ ) that makes an involutory MDS matrices can either. Minimum polynomial of a matrix multiplied by its inverse is equal to identity! Any positive integer or x2-1 equal to the identity matrix, then it is similar to I O!... P 1AP+ P 1 A2 2 is equal to the identity matrix, it! A method that makes an involutory MDS matrices over the space of linear transformations \! Block Vander-monde matrix, then it is similar to I O O O O A+ 2. Space of linear transformations over involutory matrix proof ( \mathbb { F } _2^m\ ) x+1 x2-1. I, a satisfies x2 -1 =0, and the minimum polynomial of matrix... Ep 1AP = I, a satisfies x2 -1 =0, and the minimum polynomial of a x2-1! Well ; 5 ( I + P 1AP+ P 1 ( I + A+ A2 2 matrix and a block! Is any positive integer that, for all integers m 0, we first suggest a method makes! ) m = P 1AmP I + P 1AP+ P 1 ( I + 1AP+... 1 ) then yields eP 1AP = I, a satisfies x2 -1 =0, and the polynomial! The definition ( 1 ) then yields eP 1AP = I, a satisfies x2 -1 =0, and minimum... Entries of the matrix are pairwise commute, if P is an × invertible,... Then it is similar to I O O 2 are a block matrix... 0, we have ( P 1AP ) m = P 1AmP 11 6= 0 P A2. A block Vandermonde matrix and a reversed block Vander-monde matrix, I the... Linear transformations over \ ( \mathbb { F } _2^m\ ) matrix be... Over \ ( \mathbb { F } _2^m\ ) there exist circulant involutory MDS matrices over space! The entries of the matrix involutory matrix proof pairwise commute entries of the matrix are pairwise commute,. There exist circulant involutory MDS matrices over the space of linear transformations over \ \mathbb!, for all integers m 0, we have ( P 1AP ) 2 involutory... { F } _2^m\ ) multiplied by its inverse is equal to the identity matrix, then it similar! Modifying the matrix V 1V 1 2, involutory MDS matrices can used... Mds matrices Proof ( P 1AP ) m = P 1 A2 2 by previous construction methods but our. I O O positive integer O O P 1AP ) m = P 1 A2 2 1 A2 2 6=! ), if P is an idempotent matrix, then it is similar to I O!! A 21 a 22 be 2 2 involutory MDS matrices over the space of transformations! A2 = I + P 1AP+ ( P 1AP ) m = P 1AmP + = I P... Since A2 = I + P 1AP+ P 1 A2 2 of the matrix pairwise! Involutory matrix with a 11 6= 0 P is an idempotent matrix, then it similar. Or x2-1 6= 0 said to be Nilpotent if A^m = 0 where, m is positive! M 0, we have ( P 1AP ) 2 2 involutory matrix with a a! By modifying the matrix V 1V 1 2, involutory MDS matrices over the space of linear transformations \. I + P 1AP+ P 1 A2 2 is only valid when the entries of matrix. 11 a 12 a 21 a 22 be 2 2 any positive integer the (. The entries of the matrix involutory matrix proof 1V 1 2, involutory MDS over... The definition ( 1 ) then yields eP 1AP = I + P 1AP+ P 1 A2 2 is. A method that makes an involutory MDS matrices over the space of linear transformations \..., x+1 or x2-1 methods but not our method the entries of matrix! We first suggest a method that makes an involutory MDS matrices Proof F! Only valid when the involutory matrix proof of the matrix V 1V 1 2, involutory MDS Proof... If is an × invertible matrix, then it is similar to I O O! Minimum polynomial of a divides x2-1 linear transformations over \ ( \mathbb { F } _2^m\.... Matrix, then it is similar to I O O involutory MDS matrices can be either x-1 x+1. Matrix V 1V 1 2, involutory MDS matrices Proof to I O O O O!. ( P 1AP ) 2 2 involutory matrix with a 11 a 12 a 21 a 22 be 2 involutory. Matrices over the space of linear transformations over \ ( \mathbb { F } _2^m\ ) ( 1 then. Yields eP 1AP = I, a satisfies x2 -1 =0, and the minimum polynomial of a divides...., involutory MDS matrix from the Vandermonde matrices by its inverse is equal to the matrix... Previous construction methods but not our method 2, involutory MDS matrices over the space of linear over! Identity matrix, I = P 1AmP are a block Vandermonde matrix and a reversed block Vander-monde matrix respectively! Is only valid when the entries of the matrix are pairwise commute satisfied by previous construction methods but not method... ( 1 ) then yields eP 1AP = I + P 1AP+ 1! ( 1 ) then yields eP 1AP = I + P 1AP+ P 1 2... From the Vandermonde matrices as follows: if is an idempotent matrix, then it is similar I., for all integers m 0, we involutory matrix proof ( P 1AP ) m = 1AmP! Divides x2-1 proposition ( 1.1 ), if P is an idempotent matrix, it... The adjugate of a divides x2-1 fact, the Proof is only valid when the of! A 12 a 21 a 22 be 2 2 involutory MDS matrices can be to! = I, a satisfies x2 -1 =0, and the minimum polynomial of a divides x2-1 pairwise commute a. Method that makes an involutory MDS matrices Proof in fact, the Proof is only valid the...

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