3A.3 Vector–Matrix Algebra

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The advantage of representing the excitations and the responses of a mechatronic system as the vectors u

and y; and the transfer functions as the matrix G is clear from the fact that the excitation – response

(input – output) equations can be expressed as the single equation

y ¼ Gu ð3A:5Þ

instead of the collection of scalar equations (Equation 3A.4).

Hence, the response vector y is obtained by premultiplying the excitation vector u by the transfer

function matrix G: Of course, certain rules of vector– matrix multiplication have to be agreed upon in

order that this single equation is consistent with the two scalar equations given by Equation 3A.4. Also,

we have to agree upon rules for the addition of vectors or matrices.

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© 2005 by Taylor & Francis Group, LLC

A vector is a special case of a matrix. Specifically, a third-order column vector is a matrix having three

rows and one column. Hence, it is a 3 £ 1 matrix. Similarly, a third-order row vector is a matrix having

one row and three columns. Accordingly, it is a 1 £ 3 matrix. It follows that we only need to know matrix

algebra; the vector algebra will follow from the results for matrices.

3A.3.1 Matrix Addition and Subtraction

Only matrices of the same size can be added. The result (sum) will also be a matrix of the same size. In

matrix addition, we add the corresponding elements (i.e., the elements at the same position) in the two

matrices, and write the results at the corresponding places in the resulting matrix.

As an example, consider the 2 £ 3 matrix

A ¼

21 0 3

2 6 22

" #

and a second matrix

B ¼

2 1 25

0 23 2

" #

The sum of these two matrices is given by

A þ B ¼

1 1 22

2 3 0

" #

The order in which the addition is done is immaterial. Hence

A þ B ¼ B þ A ð3A:6Þ

In other words, matrix addition is commutative.

Matrix subtraction is defined just like matrix addition, except the corresponding elements are

subtracted. An example is given below:

21 2

3 0

24 1

2

664

3

775

2

4 2

2 21

23 0

2

664

3

775

¼

25 0

1 1

21 1

2

664

3

775

3A.3.2 Null Matrix

The null matrix is a matrix whose elements are all zeros. Hence, when we add a null matrix to an arbitrary

matrix, the result is equal to the original matrix. We can define a null vector in a similar manner. We

can write

A þ 0 ¼ A ð3A:7Þ

As an example, the 2 £ 2 null matrix is

0 0

0 0

" #

3A.3.3 Matrix Multiplication

Consider the product AB of the two matrices A and B. Let us write this as

C ¼ AB ð3A:8Þ

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We say that B is premultiplied by A or, equivalently, A is postmultiplied by B. For this multiplication to

be possible, the number of columns in A has to be equal to the number of rows in B. Then, the number of

rows of the product matrix C is equal to the number of rows in A, and the number of columns in C is

equal to the number of columns in B.

The actual multiplication is done by multiplying the elements in a given row (say, the ith row) of A by

the corresponding elements in a given column (say, the jth column) of B and summing these products.

The result is the element cij of the product matrix C. Note that cij denotes the element that is common to

the ith row and the jth column of matrix C. So, we have

cij ¼

X

k

aikbkj ð3A:9Þ

As an example, suppose

A ¼

1 2 21

3 23 4

" #

B ¼

1 21 2 4

2 3 24 2

5 23 1 0

2

664

3

775

Note that the number of columns in A is equal to three and the number of rows in B is also equal to three.

Hence, we can perform the premultiplication of B by A. For example

c11 ¼ 1 £ 1 þ 2 £ 2 þ ð21Þ £ 5 ¼ 0

c12 ¼ 1 £ ð21Þ þ 2 £ 3 þ ð21Þ £ ð23Þ ¼ 8

c13 ¼ 1 £ 2 þ 2 £ ð24Þ þ ð21Þ £ 1 ¼ 27

c14 ¼ 1 £ 4 þ 2 £ 2 þ ð21Þ £ 0 ¼ 8

c21 ¼ 3 £ 1 þ ð23Þ £ 2 þ 4 £ 5 ¼ 17

c22 ¼ 3 £ ð21Þ þ ð23Þ £ 3 þ 4 £ ð23Þ ¼ 224

and so on. The product matrix is

C ¼

0 8 27 8

17 224 22 6

" #

It should be noted that both products AB and BA are not always defined and, even when they are

defined, the two results are not equal in general. Unless both A and B are square matrices of the same

order, the two product matrices will not be of the same order.

Summarizing, matrix multiplication is not commutative:

AB – BA ð3A:10Þ

3A.3.4 Identity Matrix

An identity matrix (or unity matrix) is a square matrix whose diagonal elements are all equal to 1 and all

the remaining elements are zeros. This matrix is denoted by I.

For example, the third-order identity matrix is

I ¼

1 0 0

0 1 0

0 0 1

2

664

3

775

Modal Analysis 3-45

© 2005 by Taylor & Francis Group, LLC

It is easy to see that when any matrix is multiplied by an identity matrix (provided, of course, that the

multiplication is possible) the product is equal to the original matrix; thus

AI ¼ IA ¼ A ð3A:11Þ

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