a vector can be interpreted as a file of data
DESCRIPTION
Short recapitulation of matrix basics. A vector can be interpreted as a file of data. A matrix is a collection of vectors and can be interpreted as a data base. The red matrix contain three column vectors. - PowerPoint PPT PresentationTRANSCRIPT
Species Taxon GuildMean length (mm)
Site 1 Site 2 Site 3 Site 4
Nanoptilium kunzei (Heer, 1841) Ptiliidae Necrophagous 0.60 0 0 0 0Acrotrichis dispar (Matthews, 1865) Ptiliidae Necrophagous 0.65 13 0 4 7Acrotrichis silvatica Rosskothen, 1935 Ptiliidae Necrophagous 0.80 16 0 2 0Acrotrichis rugulosa Rosskothen, 1935 Ptiliidae Necrophagous 0.90 0 0 1 0Acrotrichis grandicollis (Mannerheim, 1844) Ptiliidae Necrophagous 0.95 1 0 0 1Acrotrichis fratercula (Matthews, 1878) Ptiliidae Necrophagous 1.00 0 1 0 0Carcinops pumilio (Erichson, 1834) Histeridae Predator 2.15 1 0 0 0Saprinus aeneus (Fabricius, 1775) Histeridae Predator 3.00 13 23 4 9Gnathoncus nannetensis (Marseul, 1862) Histeridae Predator 3.10 0 0 0 2Margarinotus carbonarius (Hoffmann, 1803) Histeridae Predator 3.60 0 5 0 0Rugilus erichsonii (Fauvel, 1867) Staphylinidae Predator 3.75 8 0 5 0Margarinotus ventralis (Marseul, 1854) Histeridae Predator 4.00 3 2 6 1Saprinus planiusculus Motschulsky, 1849 Histeridae Predator 4.45 0 5 0 0Margarinotus merdarius (Hoffmann, 1803) Histeridae Predator 4.50 5 0 6 0
A vector can be interpreted as a
file of data
A matrix is a collection of
vectors and can be interpreted as a data base
The red matrix contain three
column vectors
Handling biological data is most easily done with a matrix approach.An Excel worksheet is a matrix.
Short recapitulation of matrix basics
11 1n
m1 mn
a aA
a a
1
2
3
4
aa
Vaa
1 2 3 4V a a a a
The first subscript denotes rows, the second columns.n and m define the dimension of a matrix. A has m rows and n columns.
Column vector
Row vector
1864875365424321
A
The symmetric matrix is a matrix where An,m = A m,n.
1000070000400001
A
The diagonal matrix is a square and symmetrical.
1000010000100001
A
Unit matrix Iis a matrix with one row and one column. It is a scalar (ordinary number).
3Λ
For a non-singular square matrix the inverse is defined as
IAAIAA
1
1
987642321
A
1296654321
A
r2=2r1 r3=2r1+r2
Singular matrices are those where some rows or columns can be expressed by a linear
combination of others.Such columns or rows do not contain additional
information.They are redundant.
nnkkkk VVVVV ...332211
A linear combination of vectors
A matrix is singular if it’s determinant is zero.
122122112221
1211
2221
1211
aaaaaaaa
Det
aaaa
AA
A
Det A: determinant of A
A matrix is singular if at least one of the parameters k is not zero.
(A•B)-1 = B-1 •A-1 ≠ A-1 •B-1
1112
2122
21122211
1
2212
2111
1aaaa
aaaa
aaaa
A
A
Determinant
The inverse of a 2x2 matrix
nmnmnn
mm
baba
baba
..............................
......
11
111111
BA
BB
nmn
m
bb
bb
..............................
......
1
111
Addition and subtraction Scalar product
m m
1i i1 1i iki 1 i 111 1m 11 1k 1 1 1 k
m mn1 nm m1 mk m 1 m k
ni i1 ni iki 1 i 1
a b ... a ba ... a b ... b A B ... A B
A B ... ... ... ... ... ... ... ... ... ... ... ...a ... a a ... a A B ... A B
a b ... a b
The inner or dot product
A B B A(A B) C A (B C) A B C(A B) C A C B C
Basic rule of matrix multiplication
izyzlmkljkij CZDCBA ...
BAXIAA
BAAXABAX
1
1
11
XXIIX
I
1...00............0...100...01
Identity matrix
Only possible if A is not singular.If A is singular the system has no solution.
The general solution of a linear system
13.25.091283310423
zyxzyxzyx
Systems with a unique solution
The number of independent equations equals the number of unknowns.
3.25.09833423
13.25.091283310423
X: Not singular
0678.05627.43819.0
11210
3.25.09833423 1
zyx
∆𝑁=𝑟𝑁 − 𝑟𝐾 𝑁 2
Species Aspilota sp2 Aspilota sp51981 3.8 0.71982 3.5 0.51983 6.6 01984 5.8 2.31985 0.8 01986 26.8 13.41987 18.3 5.8
Aspilota sp2 Aspilota sp5DN N -N2 DN N -N2
-0.3 3.8 -14.44 -0.2 0.7 -0.493.1 3.5 -12.25 -0.5 0.5 -0.25-0.8 6.6 -43.56 2.3 0 0-5 5.8 -33.64 -2.3 2.3 -5.2926 0.8 -0.64 13.4 0 0
-8.5 26.8 -718.24 -7.6 13.4 -179.56
𝑌=𝑋𝑎𝑋𝑇𝑌=𝑋𝑇 𝑋𝑎
=IA=A
Transpose3.8 3.5 6.6 5.8 0.8 26.8
-14.44 -12.25 -43.56 -33.64 -0.64 -718.24
XTX822.77 -19829.7
-19829.7 519256.8
(XTX)-1
0.015267 0.0005830.000583 2.42E-05
XTY-231.57
6257.805
r K0.11308 6.90785
r/K 0.01637
r K-1.0019 30.9025
r/K -0.03242
Aspilota sp2
Aspilota sp5
Both species have low reproductive rate r. They are prone to fast extinction.
The general solution of a linear system
Orthogonal vectors
X=
Y= XY= The dot product of two orthogonal vectors is zero.
If the orthogonal vectors have unity length they are called orthonormal.
A system of n orthogonal vectors spans an n-dimensional hypervolume (a Cartesian system)
In ecological modelling orthogonal vectors are of particular importance. They define linearly independent variables.
Orthogonal matrix𝐴=( 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼
−𝑠𝑖𝑛𝛼 𝑐 𝑜𝑠𝛼)𝐴𝑇=(𝑐𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛼 𝑐 𝑜𝑠𝛼 )
𝐴′ 𝐴=(𝑐 𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼 )( 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼
−𝑠𝑖𝑛𝛼 𝑐 𝑜𝑠𝛼)Multiplying an orthogonal matrix with its transpose gives the identity matrix.
𝐴−1=𝐴𝑇
The transpose of an orthogonal system is identical to its inverse.
𝐴𝑇 𝐴=(1 00 1)𝐴−1 𝐴=(1 0
0 1)
d=1Y=sin(a)
X=cos(a)
V=
X
Y How to transform vector A into vector B?
A
B
BXA
97
31
5.25.121
Multiplication of a vector with a square matrix defines a new
vector that points to a different direction. The matrix defines a
transformation in space
X
Y
A
B
BXA
Image transformationX contains all the information
necesssary to transform the image
The vectors that don’t change during transformation are the
eigenvectors.
In general we define
U is the eigenvector and the eigenvalue of the square matrix X
Eigenvalues and eigenvectors
𝑿𝑨=𝑨
𝑿𝑼=𝝀𝑼
[X [X
A matrix with n columns has n eigenvalues and n eigenvectors.
Some properties of eigenvectors
11
UUAAUUUΛAUUΛΛU
If is the diagonal matrix of eigenvalues:
The product of all eigenvalues equals the
determinant of a matrix.
n
i i1det A
The determinant is zero if at least one of the eigenvalues is zero.
In this case the matrix is singular.
The eigenvectors of symmetric matrices are orthogonal
0':)(
UUA symmetric
Eigenvectors do not change after a matrix is multiplied by a scalar k.
Eigenvalues are also multiplied by k.
0][][ uIkkAuIA
If A is trianagular or diagonal the eigenvalues of A are the diagonal
entries of A.A Eigenvalues
2 3 -1 3 23 2 -6 3
4 -5 45 5
A B C D EA 1 2 3 4 5B 2 1 4 3 2C 3 4 1 3 4D 4 3 3 1 4E 5 2 4 4 1
A -4.37578B -3.49099C -2.20138D 0.347457E 14.72069
Eigenvalues of M
A B C D EA 0.438984 0.629065 0.007298 0.443962 0.46305B -0.29098 0.442618 -0.25089 -0.72095 0.369735C 0.435137 -0.37779 0.56955 -0.37455 0.450844D 0.127886 -0.50284 -0.71251 0.121699 0.456425E -0.71898 -0.11313 0.323958 0.357809 0.487132
A B C D EA 1 1.39E-17 6.11E-16 3.89E-16 1.22E-15B 1.39E-17 1 2.29E-16 -3.5E-16 1.25E-16C 6.11E-16 2.29E-16 1 1.39E-16 -5E-16D 3.89E-16 -3.5E-16 1.39E-16 1 -6.1E-16E 1.22E-15 1.25E-16 -5E-16 -6.1E-16 1
Matrix M
Eigenvectors U of M
UTU
UTU = I
The largest eigenvalue is associated with the left (dominant) eigenvector
𝑿𝑼=𝝀𝑼
0
2
4
6
8
10
0 2 4 6 8 10
Y
X
X Y7.492729 8.2992473.794709 6.6688417.188977 4.6550685.192209 10.101633.358493 3.7593260.543067 0.145558.105676 9.837813.094105 4.8852977.392673 4.355692.225443 1.0447799.748683 6.696282.831838 1.5910568.602463 6.5004772.977185 4.208492
3.5781 5.6976052.730209 1.4998517.122361 8.5626525.771215 7.3545062.740751 1.5327675.741111 2.2855760.301084 0.052058
X YX 1 0.7218Y 0.7218 1
Correlation matrixEigenvalues
0.27821.7218
EV1 EV20.707 0.707-0.707 0.707
Xmean
Ymean 12
The eigenvectors define the major axes of the data.The eigenvalues define the length of the eigenvalues
A geometrical interpretation of eigenvalues
(1 𝑟𝑟 1)−(𝜆1 0
0 𝜆1)=0
`
(𝜆1−1)2=𝑟2
𝜆1=+𝑟+1
X YX 1 0.7218Y 0.7218 1
Correlation matrixEigenvalues
0.27821.7218
[R
The eigenvalues of a correlation similarity matrix are linearly linked to the coefficients of
correlation.
0
2
4
6
8
10
0 2 4 6 8 10
Y
X
Xmean
The eigenvector ellipse
𝐴=𝜋 𝜆1 𝜆2=𝜋(1+𝑟 )(1−𝑟 )
𝜆2=−𝑟 +1
Eigenvectors and information content
𝑿𝑼=𝝀𝑼
A matrix is a data base that contains an amount of information.
Left and right sides of an equation contain the
same amount of information
The eigenvectors take over the information content of the data base (the matrix)
The eigenvalues define ow much information contains each eigenvector.The eigenvalue is a measure of correlation.
The squared eigenvalue is therefore a measure of the variance explained by the associated eigenvector.
The eigenvector of the largest eigenvalue is called the dominant eigenvector and contains the largest part of information of the associated data base.