Bit Matrices

In document Linear Algebra (Page 154-159)

7.5 Inverse Matrix

7.5.5 Bit Matrices

In computer science, information is recorded using binary strings of data. For example, the following string contains an English word:

011011000110100101101110011001010110000101110010

A bit is the basic unit of information, keeping track of a single one or zero. Computers can add and multiply individual bits very quickly.

In chapter5, section5.2it is explained how to formulate vector spaces over

fieldsother than real numbers. In particular, al of the properties of a vector space make sense with numbersZ2 ={0,1}with addition and multiplication given by the following tables.

+ 0 1 0 0 1 1 1 0 × 0 1 0 0 0 1 0 1

7.6 Review Problems 155

Notice that−1 = 1, since 1 + 1 = 0. Therefore, we can apply all of the linear algebra we have learned thus far to matrices with Z2 entries. A matrix with entries in Z2 is sometimes called a bit matrix.

Example 95   1 0 1 0 1 1 1 1 1 

 is an invertible matrix overZ2;   1 0 1 0 1 1 1 1 1   −1 =   0 1 1 1 0 1 1 1 1  .

This can be easily verified by multiplying:

  1 0 1 0 1 1 1 1 1     0 1 1 1 0 1 1 1 1  =   1 0 0 0 1 0 0 0 1  

Application: Cryptography A very simple way to hide information is to use a sub- stitution cipher, in which the alphabet is permuted and each letter in a message is systematically exchanged for another. For example, the ROT-13 cypher just exchanges a letter with the letter thirteen places before or after it in the alphabet. For example, HELLO becomes URYYB. Applying the algorithm again decodes the message, turning URYYB back into HELLO. Substitution ciphers are easy to break, but the basic idea can be extended to create cryptographic systems that are practically uncrackable. For example, a one-time pad is a system that uses a different substitution for each letter in the message. So long as a particular set of substitutions is not used on more than one message, the one-time pad is unbreakable.

English characters are often stored in computers in the ASCII format. In ASCII, a single character is represented by a string of eight bits, which we can consider as a vector in Z82 (which is like vectors in R8, where the entries are zeros and ones). One

way to create a substitution cipher, then, is to choose an 8×8 invertible bit matrix

M, and multiply each letter of the message byM. Then to decode the message, each string of eight characters would be multiplied by M−1.

To make the message a bit tougher to decode, one could consider pairs (or longer sequences) of letters as a single vector in Z162 (or a higher-dimensional space), and

then use an appropriately-sized invertible matrix. For more on cryptography, see “The Code Book,” by Simon Singh (1999, Doubleday).

Review Problems

1. Find formulas for the inverses of the following matrices, when they are not singular: (a)   1 a b 0 1 c 0 0 1   (b)   a b c 0 d e 0 0 f  

When are these matrices singular?

2. Write down all 2×2 bit matrices and decide which of them are singular. For those which are not singular, pair them with their inverse.

3. Let M be a square matrix. Explain why the following statements are equivalent:

(a) M X =V has a unique solution for every column vector V. (b) M is non-singular.

Hint: In general for problems like this, think about the key words: First, suppose that there is some column vectorV such that the equa- tion M X = V has two distinct solutions. Show that M must be sin- gular; that is, show thatM can have no inverse.

Next, suppose that there is some column vectorV such that the equa- tion M X =V has no solutions. Show that M must be singular. Finally, suppose that M is non-singular. Show that no matter what the column vectorV is, there is a unique solution to M X =V.

Hint

4. Left and Right Inverses: So far we have only talked about inverses of square matrices. This problem will explore the notion of a left and right inverse for a matrix that is not square. Let

A=

0 1 1 1 1 0

7.6 Review Problems 157 (a) Compute: i. AAT, ii. AAT−1 , iii. B :=AT AAT−1

(b) Show that the matrixB above is aright inversefor A, i.e., verify that

AB=I .

(c) IsBA defined? (Why or why not?)

(d) Let A be an n×m matrix with n > m. Suggest a formula for a left inverse C such that

CA=I

Hint: you may assume that ATA has an inverse.

(e) Test your proposal for a left inverse for the simple example

A= 1 2 ,

(f) True or false: Left and right inverses are unique. If false give a counterexample.

Hint

5. Show that if the range (remember that the range of a function is the set of all its outputs, not the codomain) of a 3×3 matrix M (viewed as a function R3 →R3) is a plane then one of the columns is a sum of multiples of the other columns. Show that this relationship is preserved under EROs. Show, further, that the solutions toM x= 0 describe this relationship between the columns.

6. If M and N are square matrices of the same size such that M−1 exists and N−1 does not exist, does (M N)−1 exist?

8. Elementary Column Operations (ECOs) can be defined in the same 3 types as EROs. Describe the 3 kinds of ECOs. Show that if maximal elimination using ECOs is performed on a square matrix and a column of zeros is obtained then that matrix is not invertible.

In document Linear Algebra (Page 154-159)