MATRIX ALGEBRA

WHAT IS A MATRIX

DEFINITION: ANY ARRAY OF NUMBERS

Examples of Matrices

Matrices

MATRICES ARE DENOTED WITH UPPER CASE LETTERS

ELEMENTS OF MATRIX ARE DENOTED WITH SUBSCRIPTED LOWER CASE LETTERS

EXAMPLE:

a₁₁ = ELEMENT IN FIRST ROW, FIRST COLUMN

ORDER OF A MATRIX

REFERS TO HOW MANY ROWS AND COLUMNS IT CONTAINS
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•ORDER OF A MATRIX IS rxc
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ORDER OF A MATRIX

IT IS USEFUL TO WRITE THE ORDER OF A MATRIX AT THE LOWER LEFT CORNER TO KEEP TRACK OF THE ORDER

VECTOR = IS A MATRIX WITH ONE ROW OR COLUMN

Vector

A VECTOR IS SYMBOLIZED BY A LOWER CASE LETTER WITH A TILDA

VECTORS ARE USEFUL FOR DESCRIBING A SYSTEM , OR ITEM WITH MORE THAN ONE CHARACTERISTIC

Vectors (examples)

INDIVIDUAL [HT WT AGE]

POPULATION [YOUNG AGE1 AGE2 AGE3]

LANDSCAPE [%FARMLAND %TREES %WETLAND]

SCALAR

A SCALAR = A MATRIX WITH JUST ONE NUMBER

MATRIX

A SCALAR IS DENOTED BY LOWER CASE LETTER

rations .html">MATRIX OPERATIONS

ADDITION

THEN A + B =

TWO MATRICES MUST HAVE SAME ORDER TO BE ADDED

Matrix Operations

SUBTRACTION== SAME AS ADDITION

MUST HAVE EQUAL ORDER

SUBTRACT CORRESPONDING ELEMENTS

MATRIX MULTIPLICATION

SCALAR MULTIPLICATION

MULTIPLY EACH ELEMENT IN MATRIX BY A SCALAR

HIGHER ORDER MULTIPLICATION

WARNING:‐‐DO NOT SIMPLY MULTIPLY LIKE ELEMENTS

MULTIPLY ELEMENTS OF EACH ROW WITH CORRESPONDING ELEMENTS IN EACH COLUMN AND ADD THEM UP

MATRIX MULTIPLICATION

RULES

COLUMNS OF FIRST MATRIX MUST EQUAL ROWS OF SECOND MATRIX

MATRIX MULTIPLICATION

EXAMPLE:

THEN:

MATRIX MULTIPLICATION

NOTE: A X B DOES NOT EQUAL B X A

MATRIX MULTIPLICATION

RULES: A: NUMBER OF COLUMNS OF FIRST MATRIX MUST EQUAL NUMBER OF ROWS OF SECOND MATRIX

WILL PRODUCE
lXn MATRIX

NOTE : ROWS OF FIRST MATRIX AN COLUMNS OF SECOND MATRIX NEED NOT MATCH

MULTIPLICATION OF MATRIX BY A VECTOR

EXAMPLE:

MULTIPLICATION OF VECTOR BY A MATRIX

PROPERTIES OF MATRIX ALGEBRA

(A+B) = (B+A) ADDITION IS COMMUTATIVE (INDEPENDENCE OF ORDER IN WHICH THE ELEMENTS ARE TAKEN)

(A‐B) DOES NOT EQUAL (B‐A)

(AxB) DOES NOT EQUAL (BxA) MULTIPLICATION IS NOT COMMUTATIVE

axB = Bxa MULTIPLICATION WITH A SCALAR IS COMMUTATIVE

LESLIE MATRIX

SO WHAT DOES ALL THIS HAVE TO DO WITH POPULATION MODELS?

MATRIX ALGEBRA IS A CONVENIENT WAY TO KEEP TRACK OF AGE STRUCTURE DURING POPULATION GROWTH

LET'S TAKE A LOOK AT HOW WE WOULD DO THIS

EXAMPLE 1:

SUPPOSE THAT MAMMAL POPULATION WITH 4 AGE CLASSES

Where: N_x = number; M_x = fecundity; S_x = SurvivalLET N_0t = NUMBER OF IN DIVIDUALS AGED 0 AT TIME t

LET N_1t = NUMBER OF INDIVIDUALS AGED 1 AT TIME t

LET N_2t = NUMBER OF INDIVIDUALS AGED 2 AT TIME t

LET N_3t = NUMBER OF INDIVIDUALS AGED 3 AT TIME t

NOTE: AS SUMES THAT POPULATION REPRODUCES, THEN DIES.

PUT THESE COEFFICIENTS IN MATRIX FORM