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4 Linear Algebra
1. Matrices . . . 4-1 2. Special Types of Matrices . . . 4-1 3. Row Equivalent Matrices . . . 4-2 4. Minors and Cofactors . . . 4-2 5. Determinants . . . 4-3 6. Matrix Algebra . . . 4-4 7. Matrix Addition and Subtraction . . . 4-4 8. Matrix Multiplication . . . 4-4 9. Transpose . . . 4-5 10. Singularity and Rank . . . 4-5 11. Classical Adjoint . . . 4-5 12. Inverse . . . 4-6 13. Writing Simultaneous Linear Equations in
Matrix Form . . . 4-6 14. Solving Simultaneous Linear Equations . . . . 4-7 15. Eigenvalues and Eigenvectors . . . 4-8
1. MATRICES
A matrix is an ordered set of entries (elements) arranged rectangularly and set off by brackets.1 The entries can be variables or numbers. A matrix by itself has no particular value—it is merely a convenient method of representing a set of numbers.
The size of a matrix is given by the number of rows and columns, and the nomenclature m n is used for a matrix with m rows and n columns. For a square matrix, the number of rows and columns will be the same, a quantity known as the order of the matrix.
Bold uppercase letters are used to represent matrices, while lowercase letters represent the entries. For example, a23would be the entry in the second row and third column of matrix A.
A ¼
a11 a12 a13
a21 a22 a23
a31 a32 a33
2 64
3 75
A submatrix is the matrix that remains when selected rows or columns are removed from the original matrix.2 For example, for matrix A, the submatrix remaining
after the second row and second column have been removed is
a11 a13
a31 a33
" #
An augmented matrix results when the original matrix is extended by repeating one or more of its rows or col-umns or by adding rows and colcol-umns from another matrix. For example, for the matrix A, the augmented matrix created by repeating the first and second col-umns is
a11 a12 a13 j a11 a12
a21 a22 a23 j a21 a22 a31 a32 a33 j a31 a32
2 64
3 75
2. SPECIAL TYPES OF MATRICES
Certain types of matrices are given special designations.
. cofactor matrix: the matrix formed when every entry is replaced by the cofactor (see Sec. 4.4) of that entry . column matrix: a matrix with only one column . complex matrix: a matrix with complex number
entries
. diagonal matrix: a square matrix with all zero entries except for the aijfor which i = j
. echelon matrix: a matrix in which the number of zeros preceding the first nonzero entry of a row increases row by row until only zero rows remain.
A row-reduced echelon matrix is an echelon matrix in which the first nonzero entry in each row is a 1 and all other entries in the columns are zero.
. identity matrix: a diagonal (square) matrix with all nonzero entries equal to 1, usually designated as I, having the property that AI = IA = A
. null matrix: the same as a zero matrix . row matrix: a matrix with only one row
. scalar matrix:3 a diagonal (square) matrix with all diagonal entries equal to some scalar k
1The term array is synonymous with matrix, although the former is more likely to be used in computer applications.
2By definition, a matrix is a submatrix of itself.
3Although the term complex matrix means a matrix with complex entries, the term scalar matrix means more than a matrix with scalar entries.
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. singular matrix: a matrix whose determinant is zero (see Sec. 4.10)
. skew symmetric matrix: a square matrix whose transpose (see Sec. 4.9) is equal to the negative of itself (i.e., A = At)
. square matrix: a matrix with the same number of rows and columns (i.e., m = n)
. symmetric(al) matrix: a square matrix whose trans-pose is equal to itself (i.e., At = A), which occurs only when aij= aji
. triangular matrix: a square matrix with zeros in all positions above or below the diagonal
. unit matrix: the same as the identity matrix . zero matrix: a matrix with all zero entries Figure 4.1 shows examples of special matrices.
3. ROW EQUIVALENT MATRICES
A matrix B is said to be row equivalent to a matrix A if it is obtained by a finite sequence of elementary row operations on A:
. interchanging the ith and jth rows
. multiplying the ith row by a nonzero scalar
. replacing the ith row by the sum of the original ith row and k times the jth row
However, two matrices that are row equivalent as defined do not necessarily have the same determinants.
(See Sec. 4.5.)
Gauss-Jordan elimination is the process of using these elementary row operations to row-reduce a matrix to echelon or row-reduced echelon forms, as illustrated in Ex. 4.8. When a matrix has been converted to a row-reduced echelon matrix, it is said to be in row canonical form. The phrases row-reduced echelon form and row canonical form are synonymous.
4. MINORS AND COFACTORS
Minors and cofactors are determinants of submatrices associated with particular entries in the original square matrix. The minor of entry aij is the determinant of a submatrix resulting from the elimination of the single row i and the single column j. For example, the minor corresponding to entry a12 in a 3 3 matrix A is the determinant of the matrix created by eliminating row 1 and column 2.
The cofactor of entry aijis the minor of aijmultiplied by either +1 or1, depending on the position of the entry.
(That is, the cofactor either exactly equals the minor or it differs only in sign.) The sign is determined according to the following positional matrix.4
þ1 1 þ1
For example, the cofactor of entry a12 in matrix A (described in Sec. 4.4) is
cofactor of a12 ¼ a21 a23
What is the cofactor corresponding to the 3 entry in the following matrix?
The minor’s submatrix is created by eliminating the row and column of the3 entry.
M ¼ 9 1
5 9
" #
Figure 4.1 Examples of Special Matrices
9 0 0 0
(a) diagonal (b) echelon (c) row-reduced echelon
(d) identity (e) scalar (f) triangular
4The sign of the cofactor aijis positive if (i + j) is even and is negative if (i + j) is odd.
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The minor is the determinant of M.
jMj ¼ ð9Þð9Þ ð5Þð1Þ ¼ 76
The sign corresponding to the3 position is negative.
Therefore, the cofactor is76.
5. DETERMINANTS
A determinant is a scalar calculated from a square matrix. The determinant of matrix A can be repre-sented as DfAg, Det(A), DA, or jAj.5 The following rules can be used to simplify the calculation of determinants.
. If A has a row or column of zeros, the determinant is zero.
. If A has two identical rows or columns, the determi-nant is zero.
. If B is obtained from A by adding a multiple of a row (column) to another row (column) in A, then jBj ¼ jAj.
. If A is triangular, the determinant is equal to the product of the diagonal entries.
. If B is obtained from A by multiplying one row or column in A by a scalar k, then jBj ¼ kjAj.
. If B is obtained from the n n matrix A by multi-plying by the scalar matrix k, thenjkAj ¼ knjAj.
. If B is obtained from A by switching two rows or columns in A, then jBj ¼ jAj.
Calculation of determinants is laborious for all but the smallest or simplest of matrices. For a 2 2 matrix, the formula used to calculate the determinant is easy to remember.
Two methods are commonly used for calculating the determinant of 3 3 matrices by hand. The first uses an augmented matrix constructed from the original matrix and the first two columns (as shown in Sec. 4.1).6 The determinant is calculated as the sum of the products in the left-to-right downward diagonals less the sum of the products in the left-to-right upward diagonals.
The second method of calculating the determinant is somewhat slower than the first for a 3 3 matrix but illustrates the method that must be used to calculate determinants of 4 4 and larger matrices. This method is known as expansion by cofactors. One row (column) is selected as the base row (column). The selection is arbi-trary, but the number of calculations required to obtain the determinant can be minimized by choosing the row (column) with the most zeros. The determinant is equal to the sum of the products of the entries in the base row (column) and their corresponding cofactors.
A ¼
Calculate the determinant of matrix A (a) by cofactor expansion, and (b) by the augmented matrix method.
A ¼
(a) Since there are no zero entries, it does not matter which row or column is chosen as the base. Choose the first column as the base.
jAj ¼ 2 1 2
5The vertical bars should not be confused with the square brackets used to set off a matrix, nor with absolute value.
6It is not actually necessary to construct the augmented matrix, but doing so helps avoid errors.
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(b)
12 –24 84 16 28 –54
72 –10
A = 72 – (–10) = 82
2 3 2
4 7 6
1 1
3 3 3
4
4 7
2
6. MATRIX ALGEBRA7
Matrix algebra differs somewhat from standard algebra.
. equality: Two matrices, A and B, are equal only if they have the same numbers of rows and columns and if all corresponding entries are equal.
. inequality: The 4 and 5 operators are not used in matrix algebra.
. commutative law of addition:
A þ B ¼ B þ A 4:7
. associative law of addition:
A þ ðB þ CÞ ¼ ðA þ BÞ þ C 4:8
. associative law of multiplication:
ðABÞC ¼ AðBCÞ 4:9
. left distributive law:
AðB þ CÞ ¼ AB þ AC 4:10
. right distributive law:
ðB þ CÞA ¼ BA þ CA 4:11
. scalar multiplication:
kðABÞ ¼ ðkAÞB ¼ AðkBÞ 4:12
Except for trivial and special cases, matrix multiplica-tion is not commutative. That is,
AB 6¼ BA
7. MATRIX ADDITION AND SUBTRACTION Addition and subtraction of two matrices is possible only if both matrices have the same numbers of rows and columns (i.e., order). They are accomplished by adding or subtracting the corresponding entries of the two matrices.
8. MATRIX MULTIPLICATION
A matrix can be multiplied by a scalar, an operation known as scalar multiplication, in which case all entries of the matrix are multiplied by that scalar. For example, for the 2 2 matrix A,
kA ¼ ka11 ka12 ka21 ka22
" #
A matrix can be multiplied by another matrix, but only if the left-hand matrix has the same number of columns as the right-hand matrix has rows. Matrix multiplica-tion occurs by multiplying the elements in each left-hand matrix row by the entries in each right-left-hand matrix column, adding the products, and placing the sum at the intersection point of the participating row and column.
Matrix division can only be accomplished by multiply-ing by the inverse of the denominator matrix. There is no specific division operation in matrix algebra.
Example 4.3
Determine the product matrix C.
C ¼ 1 4 3
5 2 6
" # 7 12
11 8 9 10 2 64
3 75
Solution
The left-hand matrix has three columns, and the right-hand matrix has three rows. Therefore, the two matrices can be multiplied.
The first row of the left-hand matrix and the first col-umn of the right-hand matrix are worked with first. The corresponding entries are multiplied, and the products are summed.
c11¼ ð1Þð7Þ þ ð4Þð11Þ þ ð3Þð9Þ ¼ 78
The intersection of the top row and left column is the entry in the upper left-hand corner of the matrix C.
The remaining entries are calculated similarly.
c12¼ ð1Þð12Þ þ ð4Þð8Þ þ ð3Þð10Þ ¼ 74 c21¼ ð5Þð7Þ þ ð2Þð11Þ þ ð6Þð9Þ ¼ 111 c22¼ ð5Þð12Þ þ ð2Þð8Þ þ ð6Þð10Þ ¼ 136 The product matrix is
C ¼ 78 74
111 136
" #
7Since matrices are used to simplify the presentation and solution of sets of linear equations, matrix algebra is also known as linear algebra.
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9. TRANSPOSE
The transpose, At, of an m n matrix A is an n m matrix constructed by taking the ith row and making it the ith column. The diagonal is unchanged. For example,
A ¼
Transpose operations have the following characteristics.
ðAtÞt¼ A 4:13
10. SINGULARITY AND RANK
A singular matrix is one whose determinant is zero.
Similarly, a nonsingular matrix is one whose determi-nant is nonzero.
The rank of a matrix is the maximum number of linearly independent row or column vectors.8A matrix has rank r if it has at least one nonsingular square submatrix of order r but has no nonsingular square submatrix of order more than r. While the submatrix must be square (in order to calculate the determinant), the original matrix need not be.
The rank of an m n matrix will be, at most, the smaller of m and n. The rank of a null matrix is zero.
The ranks of a matrix and its transpose are the same. If a matrix is in echelon form, the rank will be equal to the number of rows containing at least one nonzero entry.
For a 3 3 matrix, the rank can either be 3 (if it is nonsingular), 2 (if any one of its 2 2 submatrices is nonsingular), 1 (if it and all 2 2 submatrices are singular), or 0 (if it is null).
The determination of rank is laborious if done by hand.
Either the matrix is reduced to echelon form by using elementary row operations, or exhaustive enumeration is used to create the submatrices and many determi-nants are calculated. If a matrix has more rows than columns and row-reduction is used, the work required to put the matrix in echelon form can be reduced by work-ing with the transpose of the original matrix.
Example 4.4
What is the rank of matrix A?
A ¼
Matrix A is singular because jAj ¼ 0. However, there is at least one 2 2 nonsingular submatrix:
1 2
Therefore, the rank is 2.
Example 4.5
Determine the rank of matrix A by reducing it to eche-lon form.
By inspection, the matrix can be row-reduced by sub-tracting two times the second row from the third row.
The matrix cannot be further reduced. Since there are two nonzero rows, the rank is 2.
7 4 9 1
The classical adjoint is the transpose of the cofactor matrix. The resulting matrix can be designated as Aadj, adjfAg, or Aadj.
Example 4.6
What is the classical adjoint of matrix A?
A ¼
8The row rank and column rank are the same.
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Solution
The matrix of cofactors is
18 2 4
The transpose of the matrix of cofactors is
Aadj¼ identity matrix, I. Only square matrices have inverses, but not all square matrices are invertible. A matrix has an inverse if and only if it is nonsingular (i.e., its deter-minant is nonzero).
AA1¼ A1A ¼ I 4:19
ðABÞ1¼ B1A1 4:20 The inverse of a 2 2 matrix is easily determined by formula.
For a 3 3 or larger matrix, the inverse is determined by dividing every entry in the classical adjoint by the determinant of the original matrix.
A1¼Aadj
jAj 4:22
Example 4.7
What is the inverse of matrix A?
A ¼ 4 5 2 3
" #
Solution
The determinant is calculated as jAj ¼ ð4Þð3Þ ð2Þð5Þ ¼ 2
Using Eq. 4.22, the inverse is
A1¼
13. WRITING SIMULTANEOUS LINEAR EQUATIONS IN MATRIX FORM
Matrices are used to simplify the presentation and solu-tion of sets of simultaneous linear equasolu-tions. For example, the following three methods of presenting simultaneous linear equations are equivalent:
a11x1þ a12x2¼ b1
In the second and third representations, A is known as the coefficient matrix, X as the variable matrix, and B as the constant matrix.
Not all systems of simultaneous equations have solu-tions, and those that do may not have unique solutions.
The existence of a solution can be determined by calcu-lating the determinant of the coefficient matrix. These rules are summarized in Table 4.1.
. If the system of linear equations is homogeneous (i.e., B is a zero matrix) and jAj is zero, there are an infinite number of solutions.
. If the system is homogeneous and jAj is nonzero, only the trivial solution exists.
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. If the system of linear equations is nonhomogeneous (i.e., B is not a zero matrix) and jAj is nonzero, there is a unique solution to the set of simultaneous equations.
. If jAj is zero, a nonhomogeneous system of simulta-neous equations may still have a solution. The requirement is that the determinants of all substitu-tional matrices (see Sec. 4.14) are zero, in which case there will be an infinite number of solutions. Other-wise, no solution exists.
14. SOLVING SIMULTANEOUS LINEAR EQUATIONS
Gauss-Jordan elimination can be used to obtain the solution to a set of simultaneous linear equations. The coefficient matrix is augmented by the constant matrix.
Then, elementary row operations are used to reduce the coefficient matrix to canonical form. All of the opera-tions performed on the coefficient matrix are performed on the constant matrix. The variable values that satisfy the simultaneous equations will be the entries in the constant matrix when the coefficient matrix is in can-onical form.
Determinants are used to calculate the solution to linear simultaneous equations through a procedure known as Cramer’s rule.
The procedure is to calculate determinants of the orig-inal coefficient matrix A and of the n matrices resulting from the systematic replacement of a column in A by the constant matrix B. For a system of three equations in three unknowns, there are three substitutional matrices, A1, A2, and A3, as well as the original coeffi-cient matrix, for a total of four matrices whose determi-nants must be calculated.
The values of the unknowns that simultaneously satisfy all of the linear equations are
x1¼jA1j
jAj 4:23
x2¼jA2j
jAj 4:24
x3¼jA3j
jAj 4:25
Example 4.8
Use Gauss-Jordan elimination to solve the following system of simultaneous equations.
2xþ 3y 4z ¼ 1 3x y 2z ¼ 4 4x 7y 6z ¼ 7 Solution
The augmented matrix is created by appending the constant matrix to the coefficient matrix.
2 3 4 j 1
3 1 2 j 4
4 7 6 j 7
2 64
3 75
Elementary row operations are used to reduce the coef-ficient matrix to canonical form. For example, two times the first row is subtracted from the third row. This step obtains the 0 needed in the a31position.
2 3 4 j 1
3 1 2 j 4
0 13 2 j 9
2 64
3 75
This process continues until the following form is obtained.
1 0 0 j 3
0 1 0 j 1
0 0 1 j 2
2 64
3 75
x = 3, y = 1, and z = 2 satisfy this system of equations.
Example 4.9
Use Cramer’s rule to solve the following system of simul-taneous equations.
2xþ 3y 4z ¼ 1 3x y 2z ¼ 4 4x 7y 6z ¼ 7
Table 4.1 Solution Existence Rules for Simultaneous Equations
B = 0 B 6¼ 0
jAj ¼ 0 infinite number of solutions (linearly dependent equations)
either an infinite number of solutions or no solution at all
jAj 6¼ 0 trivial solution only (xi= 0)
unique nonzero solution
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Solution
The determinant of the coefficient matrix is
jAj ¼
The determinants of the substitutional matrices are
jA1j ¼
The values of x, y, and z that will satisfy the linear equations are
15. EIGENVALUES AND EIGENVECTORS Eigenvalues and eigenvectors (also known as character-istic values and charactercharacter-istic vectors) of a square matrix A are the scalars k and matrices X such that
AX ¼ kX 4:26
The scalar k is an eigenvalue of A if and only if the matrix (kI A) is singular; that is, if jkI Aj ¼ 0. This equation is called the characteristic equation of the matrix A. When expanded, the determinant is called the characteristic polynomial. The method of using the characteristic polynomial to find eigenvalues and eigen-vectors is illustrated in Ex. 4.10.
If all of the eigenvalues are unique (i.e., nonrepeating), then Eq. 4.27 is valid.
½kI AX ¼ 0 4:27
Example 4.10
Find the eigenvalues and nonzero eigenvectors of the matrix A.
The characteristic polynomial is found by setting the determinantjkI Aj equal to zero.
ðk 2Þðk 4Þ ð6Þð4Þ ¼ 0 k2 6k 16 ¼ ðk 8Þðk þ 2Þ ¼ 0
The roots of the characteristic polynomial are k = +8 and k =2. These are the eigenvalues of A.
Substituting k = 8,
kI A ¼ 8 2 4
The resulting system can be interpreted as the linear equation 6x1 4x2= 0. The values of x that satisfy this equation define the eigenvector. An eigenvector X asso-ciated with the eigenvalue +8 is
X ¼ x1
All other eigenvectors for this eigenvalue are multiples of X. Normally X is reduced to smallest integers.
X ¼ 2 3
" #
Similarly, the eigenvector associated with the eigen-value2 is
Reducing this to smallest integers gives
X ¼ þ1
1
" #
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5 Vectors
1. Introduction . . . 5-1 2. Vectors in n-Space . . . 5-1 3. Unit Vectors . . . 5-2 4. Vector Representation . . . 5-2 5. Conversion Between Systems . . . 5-2 6. Vector Addition . . . 5-3 7. Multiplication by a Scalar . . . 5-3 8. Vector Dot Product . . . 5-3 9. Vector Cross Product . . . 5-4 10. Mixed Triple Product . . . 5-5 11. Vector Triple Product . . . 5-5 12. Vector Functions . . . 5-5
1. INTRODUCTION
A physical property or quantity can be described by a scalar, vector, or tensor. A scalar has only magnitude.
Knowing its value is sufficient to define a scalar. Mass, enthalpy, density, and speed are examples of scalars.
Force, momentum, displacement, and velocity are exam-ples of vectors. A vector is a directed straight line with a specific magnitude and is specified completely by its direction (consisting of the vector’s angular orientation and its sense) and magnitude. A vector’s point of appli-cation (terminal point) is not needed to define the vec-tor.1Two vectors with the same direction and magnitude are said to be equal vectors even though their lines of action may be different.2
A vector can be designated by a boldface variable (as in this book) or as a combination of the variable and some other symbol. For example, the notations V, V , ^V , ~V , and V are used by different authorities to represent vectors. In this book, the magnitude of a vector can be designated by eitherjVj or V (italic but not bold).
Stress, dielectric constant, and magnetic susceptibility are examples of tensors. A tensor has magnitude in a
Stress, dielectric constant, and magnetic susceptibility are examples of tensors. A tensor has magnitude in a