ch13

Matrix Algebra

Def: Algebra [Arabic, Al-Jabr ] Bone setting

Goals: To become adept with Matrix notation, Matrix Algebra, and proving basic algebraic rules concerning Matrices.

Previously, we represented systems of linear equations by using a matrix representation. Now we further develop the concept of matrices. We consider them abstractly as mathematical objects. We begin with matrices whose entries are scalars, that is elements from a field. Unless otherwise stated, scalars will be from the field of real numbers, R. In this course, the only fields we will be concerned with are the real numbers R, the complex numbers C, and the "clock-arithmetic" numbers Z_p. With this in mind we have the following:

Definition: A Matrix over a field F is a rectangular array of scalars.

I. Matrix Notation & Terminology

If A is an mxn matrix, then the elements,

, of A are indexed by row-column subscripts.

Akamai: The subscripts

are always in row i, column j order. Usually the comma is omitted, thus

is "

-one-one" not"

-eleven".

What does

denote?

The dimension of a matrix is the number of its rows by columns, r x c. An nxn matrix is called square. Generally, we refer to matrices by using capital letters, the main exception is when the matrix is a 1xn or a mx1 matrix, in which case we use small-case letters. These special type of matrices are called vectors.

The 1x3 matrix

is a row vector.

The 3x1 matrix

is a column vector.

A solution to a system of m linear equations in n unknowns is an n-tuple of scalars. Since n-tuples may be represented in matrix form as either row or column vectors, we refer to a solution of a system as a solution vector .

The points in Euclidean n-Space Rⁿ may also be written as vectors whose entries are simply the coordinates of the point.

Sometimes a matrix is written a a set of column vectors. If the columns of a matrix A are written as a set of n column vectors

, we write

then

If we need to refer to a particular row of A, say the ith row we use the notation:

a( i, : )

Thus the second row of A is a( 2, :) = ( a₂₁ a₂₂ a₂₃ ... a_2n)

Likewise, the columns of A are referred to as either a_j or a( :, j)

The third column of A is

a( :, 3 ) =

We are now ready to define some arithmetic properties for matrices.

II. Matrix Algebra

Equality

Two matrices A and B are equal if they have the same dimension and

and

then

and

then

The Zero Matrix O

The zero matrix is any mxn matrix with entires all 0. The 2 x 4 zero matrix is:

Matrix Addition

The sum of two matrices A and B of the same dimension is:

and

then

Scalar Multiplication

Let c be a scalar from a field F, A a matrix, then

and

then

Matrix Multiplication

Let A be an mxn matrix and B be an nxp then the matrix C = AB has dimension mxp and is computed by

Akamai: Sigma notation should always be written out until you understand what it says.

This is the dot product of row i of A, with column j of B.

For example, the

element of matrix C is the sum of the products of the 4th row of A with the 3rd column of B.

1) For the above matrices, what row*column expression is equal to

2) If

and

Compute

3) If

Compute

4) Suppose you have the set M^nxn of all nxn matrices. Does this set with the definitions of addition and matrix multiplication form a field?

5) Show that if the matrix product AB is defined, then AB = (Ab₁, Ab₂, ... , Ab_n) where the b_i are the column vectors of B.

III. Matrix Equations

Once the algebraic properties of matrices have been defined, we can then write matrix expressions and equations. Two important matrix equations are:

1. The Ax = b Form

At first, matrix multiplication seems bizarre. To get some idea as to why matrix multiplication is defined the way it is,

Let

Write out

What does it look like?

2. The Ax = x₁A₁+...+x_nA_nForm

This matrix form is equivalent to the sum of n vectors. For now just observe that we can break up Ax into:

Thinking of each of the

as scalars, we can write:

Since the

are column vectors, the last equation shows that the product Ax is a linear combination of the column vectors of A. With this is mind, we might ask, when does Ax = b fail to have a solution? Answer: When b cannot be written as a linear combination of the columns of A.

Write out the following linear system in both matrix forms.

IV. Algebraic Rules

If we keep the order of operations for matrices the same as for ordinary arithmetic; parentheses, exponents, multiplication (both matrix and scalar) and addition, as a consequence of the previous definitions we have the following algebraic rules.

Theorem: If a, b are scalars, then for any matrices A, B and C, the following operations (if defined) hold:
1. A + B = B + A
2. (A + B) + C = A + (B + C)
3. (AB) C = A (BC)
4. A (B + C) = AB + AC
5. (A + B) C = AC + BC
6. (a b) A = a (bA)
7. a (AB) = (aA) B
8. (a + b) A = aA + bA
9. a (A + B) = aA + aB.

Write out the Proof of (6). The proof is direct element-wise.

Proof Of (4) The Right Distributive Property for Matrices

Assume that

is an mxn matrix and

and

are
both nxr matrices. Let

and

. We compute

and

and observe

.

D

(write this out!) Regroup,

.

but,
E

and

. q.e.d.

Which of the above algebraic rules allows us to write the powers of a Matrix as A^k ?

V. Some Special Matrices and Terminology

1. The Powers of a Matrix

Repeated multiplication of a matrix by itself is denoted by
the matrix with an integer exponent. i.e. A³.

2. The Identity Matrix I

For any real number

Similarly, for any nxn matirx there is a matrix I with the property that

For

The identity matrix is formally defined using Kronecker's d -function.

Formal Definition: The Identity Matrix is the nxn matrix

As you move around an identity matrix, the d -function takes on the values of 0 or 1.

Akamai: The column vectors of I are the standard vectors used to define a coordinate system in Euclidean n-space. (i, j, k for R³). The standard notation for the jth column vector of I is

. This the nxn identity matrix can be written as

3. A Permutation Matrix P

A Permutation Matrix P is formed by reordering the column vectors of I, the identity matrix.

is a 3x3 permutation matrix.

then

4. Matrix Inversion

Once the identity matrix is defined, we can talk about "matrix division", as I is the multiplicative identity for matrix multiplication. In the case of matrices, we look for a muliplicative inverse, A B = I.

What is the multiplicative inverse of a real number?

number

inverse

number x inverse =

Do all numbers have an inverse?

Definition: An nxn matrix A is said to be nonsingular or invertible if there exists a matrix denoted A^-1such that AA^-1 = A^-1A = I . The matrix A^-1is called the multiplicative inverse of A.

Multiply A and B. Is B the multiplicative inverse of A?

Akamai: Not all matrices have inverses.

An nxn matrix that does not have an inverse is called singular or noninvertible. If a matrix is not a square matrix, it still may have a left or right inverse.

5. The Transpose of a Matrix

The transpose of a matrix A is denoted by A^T.

a) If