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%9$Refresher:Vector and Matrix Algebra%%0RDepartment of Chemical Engineering
FAMU/FSU College of Engineering
August 15, 2002SSOutlineBasics:
Operations on vectors and matrices
Linear systems of algebraic equations
Gauss elimination
Matrix rank, existence of a solution
Inverse of a matrix
Determinants
Eigenvalues and Eigenvectors
applications
diagonalization
more
Z#Z&ZXZZZZZZ#&XOutline cont Special matrix properties
symmetric, skewsymmetric, and orthogonal matrices
Hermitian, skewHermitian, and unitary matrices(ccMatricesA matrix is a rectangular array of numbers (or functions).
The matrix shown above is of size mxn. Note that this designates first the number of rows, then the number of columns.
The elements of a matrix, here represented by the letter a with subscripts, can consist of numbers, variables, or functions of variables.>>dVectorsA vector is simply a matrix with either one row or one column. A matrix with one row is called a row vector, and a matrix with one column is called a column vector.
Transpose: A row vector can be changed into a column vector and viceversa by taking the transpose of that vector. e.g.:
P>
Matrix AdditionMatrix addition is only possible between two matrices which have the same size.
The operation is done simply by adding the corresponding elements. e.g.:
Matrix scalar multiplicationL
Multiplication of a matrix or a vector by a scalar is also straightforward:Transpose of a matrix0
Taking the transpose of a matrix is similar to that of a vector:
The diagonal elements in the matrix are unaffected, but the other elements are switched. A matrix which is the same as its own transpose is called symmetric, and one which is the negative of its own transpose is called skewsymmetric.T0 ?Matrix MultiplicationThe multiplication of a matrix into another matrix not possible for all matrices, and the operation is not commutative:
AB `" BA in general
In order to multiply two matrices, the first matrix must have the same number of columns as the second matrix has rows.
So, if one wants to solve for C=AB, then the matrix A must have as many columns as the matrix B has rows.
The resulting matrix C will have the same number of rows as did A and the same number of columns as did B.
@xNgh Matrix MultiplicationMThe operation is done as follows:
using index notation:
for example:
L#:
Linear systems of equations,One of the most important application of matrices is for solving linear systems of equations which appear in many different problems including electrical networks, statistics, and numerical methods for differential equations.
A linear system of equations can be written:
a11x1 + & + a1nxn = b1
a21x1 + & + a2nxn = b2
:
am1x1 + & + amnxn = bm
This is a system of m equations and n unknowns.
ZTZ2ZZ
5`5(Linear systems cont The system of equations shown on the previous slide can be written more compactly as a matrix equation:
Ax=b
where the matrix A contains all the coefficients of the unknown variables from the LHS, x is the vector of unknowns, and b a vector containing the numbers from the RHS
@h lGauss eliminationAlthough these types of problems can be solved easily using a wide number of computational packages, the principle of Gaussian elimination should be understood.
The principle is to successively eliminate variables from the equations until the system is in triangular form, that is, the matrix A will contain all zeros below the diagonal.TS
.Gauss elimination cont A very simple example:
x + 2y = 4
3x + 4y =38
first, divide the second equation by 2, then add to the first equation to eliminate y; the resulting system is:
x + 2y = 4
2.5x = 15 x = 6
y = 5
6Matrix rank\The rank of a matrix is simply the number of independent row vectors in that matrix.
The transpose of a matrix has the same rank as the original matrix.
To find the rank of a matrix by hand, use Gauss elimination and the linearly dependant row vectors will fall out, leaving only the linearly independent vectors, the number of which is the rank.
\]Matrix inverse
The inverse of the matrix A is denoted as A1
By definition, AA1 = A1A = I, where I is the identity matrix.
Theorem: The inverse of an nxn matrix A exists if and only if the rank A = n.
GaussJordan elimination can be used to find the inverse of a matrix by hand. j+
CDeterminantsDeterminants are useful in eigenvalue problems and differential equations.
Can be found only for square matrices.
Simple example: 2nd order determinant*
s3rd order determinant$The determinant of a 3X3 matrix is found as follows:
The terms on the RHS can be evaluated as shown for a 2nd order determinant.
BW!Some theorems for determinants(4Cramer s: If the determinant of a system of n equations with n unknowns is nonzero, that system has precisely one solution.
det(AB)=det(BA)=det(A)det(B)
>}"Eigenvalues and EigenvectorsLet A be an nxn matrix and consider the vector equation:
Ax = lx
A value of l for which this equation has a solution x`"0 is called an eigenvalue of the matrix A.
The corresponding solutions x are called the eigenvectors of the matrix A.
V9B
^#Solving for eigenvaluesAx=lx
Ax  lx = 0
(A lI)x = 0
This is a homogeneous linear system, homogeneous meaning that the RHS are all zeros.
For such a system, a theorem states that a solution exists given that det(A lI)=0.
The eigenvalues are found by solving the above equation.
j
8,D$:Solving for eigenvalues cont Simple example: find the eigenvalues for the matrix:
Eigenvalues are given by the equation det(AlI) = 0:
So, the roots of the last equation are 1 and 6. These are the eigenvalues of matrix A.
:bK\f'EigenvectorsFor each eigenvalue, l, there is a corresponding eigenvector, x.
This vector can be found by substituting one of the eigenvalues back into the original equation: Ax = lx : for the example: 5x1 + 2x2 = lx1
2x1 2x2 = lx2
Using l=1, we get x2 = 2x1, and by arbitrarily choosing x1 = 1, the eigenvector corresponding to l=1 is:
and similarly, JZZlZZE
P*Special matricesA matrix is called symmetric if:
AT = A
A skewsymmetric matrix is one for which:
AT = A
An orthogonal matrix is one whose transpose is also its inverse:
AT = A1!*
A
5
:+Complex matrices8If a matrix contains complex (imaginary) elements, it is often useful to take its complex conjugate. The notation used for the complex conjugate of a matrix A is: ,
Some special complex matrices are as follows:
Hermitian: ,T = A
SkewHermitian: ,T = A
Unitary: ,T = A1
IR>0, ",
Questions?
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%9$Refresher:Vector and Matrix Algebra%%0TMike Kirkpatrick
Department of Chemical Engineering
FAMUFSU College of Engineering
UU$DOutlineBasics:
Operations on vectors and matrices
Linear systems of algebraic equations
Gauss elimination
Matrix rank, existence of a solution
Inverse of a matrix
Determinants
Eigenvalues and Eigenvectors
applications
diagonalization
more
Z#Z&ZXZZZZZZ#&XOutline cont Special matrix properties
symmetric, skewsymmetric, and orthogonal matrices
Hermitian, skewHermitian, and unitary matrices(ccMatricesA matrix is a rectangular array of numbers (or functions).
The matrix shown above is of size mxn. Note that this designates first the number of rows, then the number of columns.
The elements of a matrix, here represented by the letter a with subscripts, can consist of numbers, variables, or functions of variables.>>dVectorsA vector is simply a matrix with either one row or one column. A matrix with one row is called a row vector, and a matrix with one column is called a column vector.
Transpose: A row vector can be changed into a column vector and viceversa by taking the transpose of that vector. e.g.:
P>
Matrix AdditionMatrix addition is only possible between two matrices which have the same size.
The operation is done simply by adding the corresponding elements. e.g.:
Matrix scalar multiplicationL
Multiplication of a matrix or a vector by a scalar is also straightforward:Transpose of a matrix0
Taking the transpose of a matrix is similar to that of a vector:
The diagonal elements in the matrix are unaffected, but the other elements are switched. A matrix which is the same as its own transpose is called symmetric, and one which is the negative of its own transpose is called skewsymmetric.T0 ?Matrix MultiplicationThe multiplication of a matrix into another matrix not possible for all matrices, and the operation is not commutative:
AB `" BA in general
In order to multiply two matrices, the first matrix must have the same number of columns as the second matrix has rows.
So, if one wants to solve for C=AB, then the matrix A must have as many columns as the matrix B has rows.
The resulting matrix C will have the same number of rows as did A and the same number of columns as did B.
@xNgh Matrix MultiplicationMThe operation is done as follows:
using index notation:
for example:
L#:
Linear systems of equations,One of the most important application of matrices is for solving linear systems of equations which appear in many different problems including electrical networks, statistics, and numerical methods for differential equations.
A linear system of equations can be written:
a11x1 + & + a1nxn = b1
a21x1 + & + a2nxn = b2
:
am1x1 + & + amnxn = bm
This is a system of m equations and n unknowns.
ZTZ2ZZ
5`5(Linear systems cont The system of equations shown on the previous slide can be written more compactly as a matrix equation:
Ax=b
where the matrix A contains all the coefficients of the unknown variables from the LHS, x is the vector of unknowns, and b a vector containing the numbers from the RHS
@h lGauss eliminationAlthough these types of problems can be solved easily using a wide number of computational packages, the principle of Gaussian elimination should be understood.
The principle is to successively eliminate variables from the equations until the system is in triangular form, that is, the matrix A will contain all zeros below the diagonal.TS
.Gauss elimination cont A very simple example:
x + 2y = 4
3x + 4y =38
first, divide the second equation by 2, then add to the first equation to eliminate y; the resulting system is:
x + 2y = 4
2.5x = 15 x = 6
y = 5
6Matrix rank\The rank of a matrix is simply the number of independent row vectors in that matrix.
The transpose of a matrix has the same rank as the original matrix.
To find the rank of a matrix by hand, use Gauss elimination and the linearly dependant row vectors will fall out, leaving only the linearly independent vectors, the number of which is the rank.
\]
!"#$Matrix inverse
The inverse of the matrix A is denoted as A1
By definition, AA1 = A1A = I, where I is the identity matrix.
Theorem: The inverse of an nxn matrix A exists if and only if the rank A = n.
GaussJordan elimination can be used to find the inverse of a matrix by hand. j+
CDeterminantsDeterminants are useful in eigenvalue problems and differential equations.
Can be found only for square matrices.
Simple example: 2nd order determinant*
s3rd order determinant$The determinant of a 3X3 matrix is found as follows:
The terms on the RHS can be evaluated as shown for a 2nd order determinant.
BW!Some theorems for determinants(4Cramer s: If the determinant of a system of n equations with n unknowns is nonzero, that system has precisely one solution.
det(AB)=det(BA)=det(A)det(B)
>}"Eigenvalues and EigenvectorsLet A be an nxn matrix and consider the vector equation:
Ax = lx
A value of l for which this equation has a solution x`"0 is called an eigenvalue of the matrix A.
The corresponding solutions x are called the eigenvectors of the matrix A.
V9B
^#Solving for eigenvaluesAx=lx
Ax  lx = 0
(A lI)x = 0
This is a homogeneous linear system, homogeneous meaning that the RHS are all zeros.
For such a system, a theorem states that a solution exists given that det(A lI)=0.
The eigenvalues are found by solving the above equation.
j
8,D$:Solving for eigenvalues cont Simple example: find the eigenvalues for the matrix:
Eigenvalues are given by the equation det(AlI) = 0:
So, the roots of the last equation are 1 and 6. These are the eigenvalues of matrix A.
:bK\f'EigenvectorsFor each eigenvalue, l, there is a corresponding eigenvector, x.
This vector can be found by substituting one of the eigenvalues back into the original equation: Ax = lx : for the example: 5x1 + 2x2 = lx1
2x1 2x2 = lx2
Using l=1, we get x2 = 2x1, and by arbitrarily choosing x1 = 1, the eigenvector corresponding to l=1 is:
and similarly, JZZlZZE
P*Special matricesA matrix is called symmetric if:
AT = A
A skewsymmetric matrix is one for which:
AT = A
An orthogonal matrix is one whose transpose is also its inverse:
AT = A1!*
A
5
:+Complex matrices8If a matrix contains complex (imaginary) elements, it is often useful to take its complex conjugate. The notation used for the complex conjugate of a matrix A is: ,
Some special complex matrices are as follows:
Hermitian: ,T = A
SkewHermitian: ,T = A
Unitary: ,T = A1
IR>0, ",
Questions?
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EigenvectorsSpecial matricesComplex matricesFonts UsedDesign TemplateEmbedded OLE Servers
Slide TitlesLE Servers
Slide Titles$Refresher:Vector and Matrix Algebra%%0TMike Kirkpatrick
Department of Chemical Engineering
FAMUFSU College of Engineering
UU$DOutlineBasics:
Operations on vectors and matrices
Linear systems of algebraic equations
Gauss elimination
Matrix rank, existence of a solution
Inverse of a matrix
Determinants
Eigenvalues and Eigenvectors
applications
diagonalization
more
Z#Z&ZXZZZZZZ#&XOutline cont Special matrix properties
symmetric, skewsymmetric, and orthogonal matrices
Hermitian, skewHermitian, and unitary matrices(ccMatricesA matrix is a rectangular array of numbers (or functions).
The matrix shown above is of size mxn. Note that this designates first the number of rows, then the number of columns.
The elements of a matrix, here represented by the letter a with subscripts, can consist of numbers, variables, or functions of variables.>>dVectorsA vector is simply a matrix with either one row or one column. A matrix with one row is called a row vector, and a matrix with one column is called a column vector.
Transpose: A row vector can be changed into a column vector and viceversa by taking the transpose of that vector. e.g.:
P>
Matrix AdditionMatrix addition is only possible between two matrices which have the same size.
The operation is done simply by adding the corresponding elements. e.g.:
Matrix scalar multiplicationL
Multiplication of a matrix or a vector by a scalar is also straightforward:Transpose of a matrix0
Taking the transpose of a matrix is similar to that of a vector:
The diagonal elements in the matrix are unaffected, but the other elements are switched. A matrix which is the same as its own transpose is called symmetric, and one which is the negative of its own transpose is called skewsymmetric.T0 ?Matrix MultiplicationThe multiplication of a matrix into another matrix not possible for all matrices, and the operation is not commutative:
AB `" BA in general
In order to multiply two matrices, the first matrix must have the same number of columns as the second matrix has rows.
So, if one wants to solve for C=AB, then the matrix A must have as many columns as the matrix B has rows.
The resulting matrix C will have the same number of rows as did A and the same number of columns as did B.
@xNgh Matrix MultiplicationMThe operation is done as follows:
using index notation:
for example:
L#:
Linear systems of equations,One of the most important application of matrices is for solving linear systems of equations which appear in many different problems including electrical networks, statistics, and numerical methods for differential equations.
A linear system of equations can be written:
a11x1 + & + a1nxn = b1
a21x1 + & + a2nxn = b2
:
am1x1 + & + amnxn = bm
This is a system of m equations and n unknowns.
ZTZ2ZZ
5`5(Linear systems cont The system of equations shown on the previous slide can be written more compactly as a matrix equation:
Ax=b
where the matrix A contains all the coefficients of the unknown variables from the LHS, x is the vector of unknowns, and b a vector containing the numbers from the RHS
@h lGauss eliminationAlthough these types of problems can be solved easily using a wide number of computational packages, the principle of Gaussian elimination should be understood.
The principle is to successively eliminate variables from the equations until the system is in triangular form, that is, the matrix A will contain all zeros below the diagonal.TS
.Gauss elimination cont A very simple example:
x + 2y = 4
3x + 4y =38
first, divide the second equation by 2, then add to the first equation to eliminate y; the resulting system is:
x + 2y = 4
2.5x = 15 x = 6
y = 5
6Matrix rank\The rank of a matrix is simply the number of independent row vectors in that matrix.
The transpose of a matrix has the same rank as the original matrix.
To find the rank of a matrix by hand, use Gauss elimination and the linearly dependant row vectors will fall out, leaving only the linearly independent vectors, the number of which is the rank.
\]Matrix inverse
The inverse of the matrix A is denoted as A1
By definition, AA1 = A1A = I, where I is the identity matrix.
Theorem: The inverse of an nxn matrix A exists if and only if the rank A = n.
GaussJordan elimination can be used to find the inverse of a matrix by hand. j+
CDeterminantsDeterminants are useful in eigenvalue problems and differential equations.
Can be found only for square matrices.
Simple example: 2nd order determinant*
s3rd order determinant$The determinant of a 3X3 matrix is found as follows:
The terms on the RHS can be evaluated as shown for a 2nd order determinant.
BW!Some theorems for determinants(4Cramer s: If the determinant of a system of n equations with n unknowns is nonzero, that system has precisely one solution.
det(AB)=det(BA)=det(A)det(B)
>}"Eigenvalues and EigenvectorsLet A be an nxn matrix and consider the vector equation:
Ax = lx
A value of l for which this equation has a solution x`"0 is called an eigenvalue of the matrix A.
The corresponding solutions x are called the eigenvectors of the matrix A.
V9B
^#Solving for eigenvaluesAx=lx
Ax  lx = 0
(A lI)x = 0
This is a homogeneous linear system, homogeneous meaning that the RHS are all zeros.
For such a system, a theorem states that a solution exists given that det(A lI)=0.
The eigenvalues are found by solving the above equation.
j
8,D$:Solving for eigenvalues cont Simple example: find the eigenvalues for the matrix:
Eigenvalues are given by the equation det(AlI) = 0:
So, the roots of the last equation are 1 and 6. These are the eigenvalues of matrix A.
:bK\f'EigenvectorsFor each eigenvalue, l, there is a corresponding eigenvector, x.
This vector can be found by substituting one of the eigenvalues back into the original equation: Ax = lx : for the example: 5x1 + 2x2 = lx1
2x1 2x2 = lx2
Using l=1, we get x2 = 2x1, and by arbitrarily choosing x1 = 1, the eigenvector corresponding to l=1 is:
and similarly, JZZlZZE
P*Special matricesA matrix is called symmetric if:
AT = A
A skewsymmetric matrix is one for which:
AT = A
An orthogonal matrix is one whose transpose is also its inverse:
AT = A1!*
A
5
:+Complex matrices8If a matrix contains complex (imaginary) elements, it is often useful to take its complex conjugate. The notation used for the complex conjugate of a matrix A is: ,
Some special complex matrices are as follows:
Hermitian: ,T = A
SkewHermitian: ,T = A
Unitary: ,T = A1
IR>0, "roK.