Matrix Algebra - OverviewMatrix Algebra - Overview

Introduction to MatricesR-mode vs. Q-modeLinear AlgebraSpecial Matrices

Trace, Diagonal, Identity, Scalars, Transpose

Matrix AdditionMatrix Multiplication

Matrix AlgebraMatrix Algebra

Matrix algebra is an essential tool for multivariate analysis because most data sets are recorded in a matrix format (rows & columns).

The use of matrices provides a succinct representation of ecological information and with matrix algebra we can perform operations and analyses of whole data sets.

We won’t dive too deeply into this in FISH 560, but it is important to have a basic understanding of matrix algebra to help grasp key concepts in multivariate statistics.

Table Structure of DataTable Structure of Data

Ecological data are typically recorded in a table, or matrix, where each column j corresponds to a descriptor yj (species abundance, habitat variable, diet item) and each row i refers to an object xi (sampling site, individual, etc.).

Objects c1 c2 c3 cj

r1 a11 a12 a13 a1j

r2 a21

r3 a31

ri ai1 aij

Each cell (i,j) of the matrix is the value of ith object for jth descriptor:

Objects and DescriptorsObjects and Descriptors

Sometimes it may not be clear as to what are the objects and what are the descriptors.

A common approach in community ecology is to compare different sampling sites (the objects) based on the assemblage of organisms observed at each one (the descriptors).

In contrast, in fish diet analyses we would consider individual fishes as objects and diet items found in fish stomachs as the descriptors.

It is therefore necessary to define what are objects and descriptors before analyzing the data.

R-mode and Q-mode anlysesR-mode and Q-mode anlyses

The distinction between objects and descriptors has very important implications.

The analysis of relationships among descriptors at the given set of objects is known as R-mode analysis.

In contrast, a Q-mode analysis quantifies the relationships among the objects given a series of descriptors.

Each type of analysis can require very different multivariate techniques

Linear AlgebraLinear Algebra

As described above, a table of ecological data is referred to as a matrix.

Linear algebra is the branch of mathematics that works with matrices.

rcr2r1

2c2221

1c1211

aaa

aaa

aaa

A

A matrix of dimension r X c is a rectangular array of numbers with r rows and c columns.

Matrix FormMatrix Form

Matrices can take many forms: rectangular, square, row or column.

ra

a

a

2

1

a

cbbbb 21

36

161A

A square matrix (r = c)

A column vector (r X 1)

A row vector (1 X c)

A scalar (1 x 1)

Matrix NotationMatrix Notation

Matrix notation provides a mechanism for writing and describing elements of data sets.

As such, it corresponds to the way computers and programming languages interpret tables of data.

Many statistical programs can handle data entered in matrix format.

So, matrices are a very useful when working with multivariate data

Square MatricesSquare Matrices

For a square matrix A (of order n × n), the diagonal elements those with identical subscripts (e.g., a11, a22, etc.). Thus, they are located on the main diagonal of the square matrix (from upper left to lower right).

nnn2n1

44

33

2n2221

1n1211

aaa

a

a

aaa

aaa

AThe sum of the diagonal elements

is known as the trace.

Diagonal MatricesDiagonal Matrices

A diagonal matrix is a square matrix in which only the diagonal elements are non-zero.

An identity matrix, I, is a diagonal matrix in which all the diagonal elements are one.

200

010

004

A

100

010

001

I

Null and Triangular MatricesNull and Triangular Matrices

A matrix where all elements are zero is called a zero matrix or null matrix. It is indicated by 0 or [0].

A square matrix with all elements below (or above) the main diagonal being zero, is called an upper (or lower) triangular matrix.

100

1880

324

A

The transpose of matrix A with dimensions (n × p) is indicated as A’ and is a new matrix of dimensions (p × n) in which a’ij = aji.

Transposing MatricesTransposing Matrices

217

4010A

24

10

710

A

Transposing matrices is very important in many forms of statistical analysis including multivariate analyses.

Symmetric MatricesSymmetric Matrices

A square matrix that is identical to its transpose is said to be symmetric. In this case, the terms aij and aji, which lie on either side of the diagonal, are equal. For example:

3183

1822

324

A

Matrix AdditionMatrix Addition

Matrices must be of the same order to be added.

If you sampled 3 sites and measured the abundance of 2 species on once per month and wanted to know the total abundance sampled, you could use matrix addition:

1015

32

1226

29

02

315

62

20

21

24

10

710

June July August Summer

Site 1

Site 2

Site 3

Sp.1

Sp.2

Matrix MultiplicationMatrix Multiplication

The result of a scalar product of two vectors is equal to the sum of the products of those corresponding order numbers. The scalar product is usually designated by a dot (or by no symbol at all). For example:

pp

p

p cbcbcb

c

c

c

bbbcbbc ...2211

2

1

21 A scalar

Going back to the earlier example, we can multiply a month matrix with a vector of appropriate order.

14

3

41

)3(2)2(4

)3(1)2(0

)3(7)2(10

3

2

24

10

710

JuneSampling efficiency

Total fish abundance

Site 1

Site 2

Site 3

Sp.1

Sp.2

Matrix Multiplication …Matrix Multiplication …

Multiplying two matrices is an extension of the product of a vector by a matrix.

To multiply matrix C by a second matrix B, consider C as a set of column vectors (e.g., c1, c2, etc.).

For example:

230

121

113

201

B

03

12

21

C edC and

Simply multiply B by the vectors d and e.

Matrix Multiplication …Matrix Multiplication …

NB: To multiply two matrices they must be conformable, which means the number of columns of the first matrix must equal the number of rows in the second.

The resulting matrix will have the same no. of rows as B and no. of columns as C.

12

8

8

7

)3(2)2(3)1(0

)3(1)2(2)1(1

)3(1)2(1)1(3

)3(2)2(0)1(1

Bd

3

4

7

2

)0(2)1(3)2(0

)0(1)1(2)2(1

)0(1)1(1)2(3

)0(2)1(0)2(1

Be

312

48

78

27

BC

and

thus