Matrix which is its own inverse

Our desired matrices satisfy the equation or. Hence the minimal polynomial of is a factor of. If it is either or this corresponds to the solutions. If the minimal polynomial is , the fact that there are no repeated factors i. Hence or where , and represents the identity matrix and is an submatrix of 0s. In the special case of this means we can also write with and and obtain. Note here that the diagonal entries sum to 0 and.

Matrices that satisfy or are known as involutory. Such matrices have applications in cryptography where it may be useful for the same matrix operation to act as its inverse in decryption. We have the following two further characterisations of involutory matrices holding true in any field of characteristic not equal to 2.

To see this, we write as before and partition and as , where is and is. Then while. The converse that if has this form then it is involutory is easier to prove: if then. As an example, since , the matrix. Levine and H. In an earlier post I had referred to involutory matrices that had the special property that they were equal to their […]. So here by their one here, by their zero Here, by their zero here, by their one.

So, Those are least three examples of matrices that are its own inverse. In mathematics, the absolute value or modulus x of a real number x is its numerical value without regard to its sign. The absolute value of a number may be thought of as its distance from zero along a number line; this interpretation is analogous to the distance function assigned to a real number in the real number system. For example, the absolute value of?

Click 'Join' if it's correct. Fred K. Algebra 3 weeks, 1 day ago. View Full Video Already have an account? Bobby B. Discussion You must be signed in to discuss. Video Transcript So one of the most um straightforward cases of this being true. Upgrade today to get a personal Numerade Expert Educator answer! Ask unlimited questions. How do we find the inverse of a matrix?

What is the multiplicative inverse of 15? Which number has no multiplicative inverse? What is the additive and multiplicative inverse of 1 2? The additive inverse and multiplicative inverse of 1 by 2 are -2 and 2 respectively. Is inverse of an element unique? What is the inverse of a relation? What is the inverse of 12? What is the negative inverse of 1? Are all square matrices symmetric? What is the inverse of a diagonal matrix?

Are Involutory matrices Diagonalizable? Show 3 more comments. Active Oldest Votes. Spenser Spenser Add a comment. Eff Eff Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events.