26. chapter 26 - remainder _a4

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REMAINDER OF SOME FUNDAMENTALS IN VECTOR AND MATRIX ALGEBRA Matrix notation was invented to express linear algebra in a compact form. Compactness enhances the visualization and understanding the essentials. Considering the following linear relations between a set of n quantities and another set of m quantities:

a11x1 + a12x2 + + a1nxn = y1 a21x1 + a22x2 + + a2nxn = y2

am1x1 + am2x2 + + amnxn = ym

they can be arranged as follows

=

mmmnmm

n

n

y

yy

x

xx

aaa

aaaaaa

.

.

.

.

.

.

.....................

...

...

2

1

2

1

21

22221

11211

The two-dimensional array is a matrix called A and the one-dimensional arrays are column vectors called x and y. Usually, to avoid brackets, a bold format is used in condensed writing:

Ax = y,

meaning a matrix product of A times x. Another advantage of the matrix notation is the computer implementation of linear algebra in programming languages that offer array data structures. A vector is a set on n numbers written in a column (column vector) or a row (row vector).

______________________Basics of the Finite Element Method Applied in Civil Engineering

279

x =

nx

xx

.2

1

, z = [ ]nzzz ...21

The numbers xi or zi are called components (or entries) while n is the order (or dimension) of the vector. Vectors of order 1 are scalars. Usually, using condensed writing, vectors are typed in bold lowercase letters

a, b, c, x, y, z or using brackets (component notation)

x = { }ix , a = { }ia , Two or three dimensional vectors have as geometrical interpretation in mechanics a position vector. A null vector, written 0, is a vector whose all components equals to zero. The unit vector, written ei, is the vector whose components equals to zero except the ith component which is one. The unitary vector, called e, is the vector with all components equal to unity. The transpose of a column vector x is the row vector that has the same components and is written xT:

xT = [ ]nxxx ...21 Transposing a vector twice yields the original vector: (xT) T = x. Two column vectors x and y of equal order n are equal (x = y) if and only if their components are equal, xi = yi, for all i = 1 n. The sum of two vectors of same order n, x + y, is defined as a new vector of order n

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x + y =

+

++

nn yx

yxyx

.

.

.22

11

the geometric interpretation for two and three-dimensional vectors being the parallelogram rule for adding. The operation is commutative and associative. The operation is undefined for vectors of different order. Multiplication of a vector x by a scalar c is defined by

cx =

ncx

cxcx

.

.

.2

1

The division of a vector by a scalar c 0 is equivalent to multiplication by 1/c. The inner product of two column vectors of same order x and y is a scalar function written (x, y) and defined by

(x, y) = xT y = yT x = in

ii yx

=1

For vectors of different order, the product in undefined. The inner product is commutative, but not generally associative: (x, (y, z)) ((x, y), z). The Euclidean norm of a real vector x (the components are real numbers) is a scalar written x that results as the vectors inner product by itself:


281

( ) =

==n

iix

1

2,xxx

Because the norm is a sum of squares, it is zero only if x is the null vector. It thus provides a meter on the vector magnitude. The Euclidean length or simply length of a real vector, written x , is the positive square root of its Euclidean norm:

xx += Cauchy Schwarz inequality: yxyx, )( Triangle inequality: yxyx ++ A vector of length one is called unit length vector. Any non null vector x can be scaled to unit length by dividing all components by its original length:

=

x

xx

xx/

/...//

2

1

nx

xx

The operation is called normalization to unit length. The angle in radians between two unit length vectors x and y, written

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cos = ( )yxyx,

yy

xx =

,

The acute angle between vector x and y spans is the real number satisfying 0 /2 and

cos = ( )yxyx,

Two vectors x and y are said to be orthogonal if (x, y) = 0. The acute angle between two orthogonal vectors = /2. The projection of vector y onto a vector x is the vector p that has the direction of x and is orthogonal to y p:

(p, y p) = 0, where p = xxc

c = cosyxxy, =

If, for a set of n-dimensional unit length vectors bk (k = 1, , m) which are mutually orthogonal, a particular n-dimensional vector x admits the representation

x = km

kkc b

=1

the coefficients ck are given by the inner products

ck = (x, bk) = kin

iibx

=1


283

The set bk is called an orthonormal basis for the vector x. The set of all vectors x forms a subspace of dimension m and the numbers ck are called the coordinates of x with respect to that basis. If n = m, set bk forms a complete orthogonal basis for the n-dimensional space. An array with m rows and n columns, obeying the following properties, is called a rectangular matrix of order m n. Conventionally, matrices are written in condensed style using bold uppercase letters (A, B, etc). The components (or entries) of a matrix have 2 indices, according to row and column number.

A = [aij] Square matrices (m = n) have some special properties, such as symmetry and antisymmetry, associated determinants and the possibility of calculating eigenvalues. The aii terms of a square matrix define the main (or cross) diagonal. Square matrices for which aii = aji are called symmetric about the main diagonal, while those for which aij = - aji are called antisymmetric (the diagonal entries of an antisymmetric matrix must be zero). The null matrix, written 0, is the matrix all of whose components are zero. The identity matrix, written I, is a square matrix all of which entries are zero except those on the main diagonal, which are ones. A diagonal matrix is a square matrix all of which entries are zero except for those on the main diagonal, which may be arbitrary. An upper triangular matrix is a square matrix in which all elements underneath the main diagonal are zero. A lower triangular matrix is a square matrix in which all elements above the main diagonal are zero. Two matrices A and B of same order m n are said to be equal if and only if all of their components are equal: aij = bij, for all i = 1, . . .m, j = 1, . . . n : A = B. Two matrices of different order can not be compared. The transpose of a matrix A is another matrix written AT that has n rows and m columns

AT = [aji]

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(AT) T = A

The transpose of a square matrix is also a square matrix. The transpose of a symmetric matrix equals the same matrix (A = AT) while the transpose of an atisymmetric matrix is the negated of the original (A = -AT). The sum of two matrices A and B is written A + B and is defined as

A + B = [aij + bij]

The addition operation is commutative and associative. To multiply a matrix by a scalar c means to multiply each entry by c

cA = [caij]

The matrix vector product, Ax = y, available only if the column dimension of the matrix A (m n) equals the dimension of the vector x (n), is a new vector y of order m, which entries are

yi = jn

jij xa

=1, i = 1, , m

The matrix-by-matrix product C of two matrices A and B:

C = AB Here A = [ai j ] is a matrix of order m n, B = [bjk ] is a matrix of order n p, and C = [cik] is a matrix of order m p. The entries of the result matrix C are defined by the formula

=

=n

jjkijik bac

1

, i = 1, , m, k = 1, , p

For product being possible, the column dimension of A must be the same as the row dimension of B (matrices are product conforming). The matrix-by-


285

matrix product is generally not commutative, even BA exists (m = n), AB BA. If A = B, the product AA is called the square of A and is denoted by A2. For this definition to make sense, A must be a square matrix. Similarly,

A3 = AAA = A2A. This definition does not encompass negative powers. The matrix-by-matrix product is associative

A(BC) = (AB)C = ABC and distributive if B and C are of the same order

A(B + C) = AB + AC and (B + C)A = BA + CA

The transpose of a matrix product is equal to the product of the transposes of the operands taken in reverse order:

(AB)T = BTAT. If B is a symmetric matrix of order m and A is an arbitrary m n matrix, then

S = ATBA is a symmetric matrix of order n. Such an operation is called a congruential transformation. It occurs very frequently in finite element analysis when changing coordinate bases because such a transformation preserves energy. The matrix product AB can be zero although A 0 and B 0. Similar, it is possible that A 0, A2 0, . . . , but Ap = 0. For x and y two column vectors of order n, and A a real square n n matrix, the following triple product produces a scalar result:

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s = yTAx which is called a bilinear form. Transposing both sides (the transpose of a scalar does not change)

s = xTATy

For A symmetric (AT = A) and x = y the bilinear form becomes a quadratic form

s = xTATx Transposing both sides of a quadratic form reproduces the same equation.

26. chapter 26 - remainder _a4

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