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Figure A.2. The physical meanings of the rows and columns of two-dimensional data matrix as generated by a hyphenated instrument.
From this perspective, vectors and matrices in linear algebra are important in mathematical manipulation of one- and two-dimensional data obtained from analytical instruments.
A.1.2. Column and Row Vectors
A group of real numbers arranged in a column form a column vector, while its transpose is a row vector as shown in the following way:
a = [a1, a2,... , an ]
Here, we follow the convention in which a boldfaced variable denotes a column vector or a matrix.
If we say two vectors, a and b, are equal to each other, that means every corresponding element in them are equal.
A.1.3. Addition and Subtraction of Vectors
Addition or subtraction of two vectors means that every element of the vectors is added or subtracted in the following manner:
a ± b =
a1 ± b a2 ± b>2
an ± bn
Vector addition and subtraction have the following properties:
a + b = b + a
(a + b) + c = a + (b + c) a + 0 = a
Here 0 = [0,0,... ,0]f.
A spectrum of a mixture of two chemical components, say, a and b, can be expressed as the vector sum of the individual spectra a and b according to the Lambert-Beer law (see Fig. A.3).
Vector addition of individual spectra to give the spectrum of the mixture can also be applied to other analytical signals such as a chromatogram, a
Figure A.3. The mixture spectrum produced by adding two spectra a and b together.
voltammogram, a kinetic curve, or a titration curve, as they are governed by additive laws similar to the Lambert-Beer law for absorbance. A vector with n elements can be regarded as a point in n-dimensional linear space. Subtraction of two vectors gives the distance between these two points the in the n-dimensional linear space. The geometric meaning of vector subtraction is shown in Figure A.4.
It is well known that addition and subtraction between vectors can be visualized by the so-called parallelogram rule as depicted in Figure A.5.
A vector in a n-dimensional linear space has direction and length. The direction of a vector is determined by the ratios between elements. The
A.1.4. Vector Direction and Length
Figure A.4. Geometric illustration of vector subtraction in an n-dimensional linear space.
Figure A.5. Parallelogram rule for vector addition and subtraction.
length or the magnitudes of a vector is defined by
||a|| = (a2 + , +a^1/2 In linear algebra, ||a|| is called the norm of the vector a.
A.1.5. Scalar Multiplication of Vectors
A vector a multiplied by a scalar (a constant) k is given by
k a =
and is called the scalar multiplication of a vector in linear algebra. Note that the spectra of different concentrations are just like vectors multiplied by different constants, say, k1, k2, and so on (see Fig. A.6).
Figure A.6. Profiles obtained by scalar multiplication of a vector (spectrum) by constants k1 and k2 with k2 > k1.
Scalar multiplication of vectors has the following properties:
k1 (k2a) = (kk2)a k1(a + b) = k1a + k1b (k1 + k2) a = ka + k2a
In particular, we have
0 a = 0 1 a = a -1 a =-a
A.1.6. Inner and Outer Products between Vectors
When two vectors with the same size (number of elements) multiply each other, there are two possible operations: the inner product and the outer product. The inner product (also known as the dot product or the scalar product) produces a scalar (a number), while the outer product (also known as the cross-product or the vector product) produces a matrix. The following formula (where the superscript t denotes transposition) defines the inner product between two vectors:
a b = [a1, a2,... , an ]
= J2 aibi
The inner product has the following properties:
af (b + c) = af b + af c (a + b)f c = af c + bf c
Figure A.7 gives the geometric meaning of the inner product between two vectors. The inner product is essentially a kind of projection. The
Inner product of vectors
b b b
Figure A.7. Graphic representation of inner product of the two vectors a and b.
concept of projection is very important in chemometrics, and a good understanding of this concept will be very helpful in studying the subject.
If two vectors a and b are orthogonal with each other, that is, if the angle, a, between them is 90° (as shown in the middle part of Fig. A.7), then the inner product is equal to zero:
af b = 0
The outer product of two vectors produces a bilinear matrix of rank equal to 1, which is of special importance in multivariate resolution for two-way data. In the two-way data from hyphenated chromatography, every chemical component can be expressed by such a bilinear matrix of rank 1. The outer product of vectors a and b is given as follows:
a1 a1 b1 a1 b2 ' ? aA"
abf = a2 a2b1 a2b2 & ro cr 3
[bi, b2,. . , bn ] =