MECH 241:
Computational Techniques
in Mechanical engineering
Linear Algebra
:
Macole Sabat, Ph.D.
Relational Operators
• Relational operators compare the elements in
two arrays and return logical true or false
values to indicate where the relation holds.
==
>=
>
<=
<
~=
isequal
isequaln
Determine equality
Determine greater than or equal to
Determine greater than
Determine less than or equal to
Determine less than
Determine inequality
Determine array equality
Determine array equality, treating NaN values as equal
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Logical Operation
• The logical data type represents true or false
states using the numbers 1 and 0, respectively.
Certain MATLAB® functions and operators
return logical values to indicate fulfillment of a
condition.
• One can use those logical values to index into
an array or execute conditional code.
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Logical Operation
Short-circuit &&, ||
&
~
|
xor
all
any
false
find
islogical
logical
true
Logical operations with short-circuiting
Find logical AND
Find logical NOT
Find logical OR
Find logical exclusive-OR
Determine if all array elements are nonzero or true
Determine if any array elements are nonzero
Logical 0 (false)
Find indices and values of nonzero elements
Determine if input is logical array
Convert numeric values to logicals
Logical 1 (true)
• With logical short-circuiting, the second operand is evaluated only
when the result is not fully determined by the first operand.
• Due to the properties of logical AND and OR, the result of a logical
expression is sometimes fully determined before evaluating all of the
conditions.
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Operators Precedence
• One can build expressions that use any
combination of arithmetic, relational, and
logical operators.
• Precedence levels determine the order in
which MATLAB® evaluates an expression.
• Within each precedence level, operators have
equal precedence and are evaluated from left
to right.
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Operators Precedence
• The precedence rules for MATLAB operators are shown in
this list, ordered from highest precedence level to lowest
precedence level:
1. Parentheses ()
2. Transpose (.'), power (.^), complex conjugate transpose
('), matrix power (^)
3. Power with unary minus (.^-), unary plus (.^+), or logical
negation (.^~) as well as matrix power with unary minus
(^-), unary plus (^+), or logical negation (^~).
Note: Although most operators work from left to right, the
operators (^-), (.^-), (^+), (.^+), (^~), and (.^~) work from
second from the right to left. It is recommended that you
use parentheses to explicitly specify the intended
precedence of statements containing these operator
combinations.
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Operators Precedence
4. Unary plus (+), unary minus (-), logical negation (~)
5. Multiplication (.*), right division (./), left division (.\),
matrix multiplication (*), matrix right division (/), matrix
left division (\)
6. Addition (+), subtraction (-)
7. Colon operator (:)
8. Less than (<), less than or equal to (<=), greater than (>),
greater than or equal to (>=), equal to (==), not equal to
(~=)
9. Element-wise AND (&)
10. Element-wise OR (|)
11. Short-circuit AND (&&)
12. Short-circuit OR (||)
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Operators Precedence
• Precedence of AND and OR Operators:
– MATLAB always gives the & operator precedence over the |
operator. Although MATLAB typically evaluates expressions
from left to right, the expression a|b&c is evaluated as
a|(b&c). It is a good idea to use parentheses to explicitly
specify the intended precedence of statements containing
combinations of & and |.
– The same precedence rule holds true for the && and ||
operators.
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Operators Precedence
• Overriding Default Precedence:
– The default precedence can be overridden using
parentheses
❖Example:
A
B
C
C
= 3;
= 2;
= A/B^2
=
0.7500
D = (A/B)^2
D =
2.2500
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Array Vs Matrix Operations
• MATLAB® has two different types of arithmetic operations:
– array operations and
– matrix operations.
• You can use these arithmetic operations to perform
numeric computations, for example, adding two numbers,
raising the elements of an array to a given power, or
multiplying two matrices.
• Matrix operations follow the rules of linear algebra.
• By contrast, array operations execute element by element
operations and support multidimensional arrays.
• The period character (.) distinguishes the array operations
from the matrix operations. However, since the matrix and
array operations are the same for addition and subtraction,
the character pairs .+ and .- are unnecessary.
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Array Operations
• Array operations execute element by element
operations on corresponding elements of
vectors, matrices, and multidimensional
arrays.
• If the operands have the same size, then each
element in the first operand gets matched up
with the element in the same location in the
second operand.
• If the operands have compatible sizes, then
each input is implicitly expanded as needed to
match the size of the other.
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Array Operations
• The following table provides a summary of arithmetic
array operators in MATLAB. For function-specific
information, click the link to the function reference
page in the last column.
Operator
+
+
.*
Purpose
Addition
Unary plus
Subtraction
Unary minus
Element-wise multiplication
.^
Element-wise power
./
Right array division
.\
Left array division
.'
Array transpose
Description
A+B adds A and B.
+A returns A.
A-B subtracts B from A
-A negates the elements of A.
A.*B is the element-by-element
product of A and B.
A.^B is the matrix with
elements A(i,j) to the B(i,j) power.
A./B is the matrix with
elements A(i,j)/B(i,j).
A.\B is the matrix with
elements B(i,j)/A(i,j).
A.' is the array transpose of A. For
complex matrices, this does not
involve conjugation.
Reference Page
plus
uplus
minus
uminus
times
power
rdivide
ldivide
transpose
❖Example: ArrayOpExp.mlx
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Note of Array Compatibility
• Most binary (two-input) operators and functions
in MATLAB® support numeric arrays that have
compatible sizes.
• Two inputs have compatible sizes if,
– for every dimension, the dimension sizes of the inputs
are either the same or one of them is 1.
• In the simplest cases, two array sizes are
compatible if they are exactly the same or if one
is a scalar.
• MATLAB implicitly expands arrays with
compatible sizes to be the same size during the
execution of the element-wise operation or
function.
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Note of Array Compatibility
• 2D inputs: combinations of scalars, vectors,
and matrices that have compatible sizes:
– Two inputs which are exactly the same size.
– One input is a scalar.
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Note of Array Compatibility
• 2D inputs: combinations of scalars, vectors,
and matrices that have compatible sizes:
– One input is a matrix, and the other is a column
vector with the same number of rows.
– One input is a column vector, and the other is a
row vector.
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Note of Array Compatibility
• 2D inputs:
• Inputs with Incompatible Sizes: Incompatible
inputs have sizes that cannot be implicitly
expanded to be the same size. For example:
– One of the dimension sizes are not equal, and neither
is 1.
A: 3-by-2
B: 4-by-2
– Two nonscalar row vectors with lengths that are not
the same.
A: 1-by-3
B: 1-by-4
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Note of Array Compatibility
• Multidimensional Arrays: Every array in MATLAB
has trailing dimensions of size 1. For
multidimensional arrays, this means that a 3-by-4
matrix is the same as a matrix of size 3-by-4-by-1by-1-by-1.
• Examples of multidimensional arrays with
compatible sizes:
– One input is a matrix, and the other is a 3-D array with
the same number of rows and columns.
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Note of Array Compatibility
• Examples of multidimensional arrays with
compatible sizes:
– One input is a matrix, and the other is a 3-D array.
The dimensions are all either the same or one of
them is 1.
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Note of Live Functions
• Live functions are program files that contain code
and formatted text together in a single interactive
environment called the Live Editor.
• Similar to live scripts, live functions allow you to
reuse sequences of commands by storing them in
program files.
• Live functions provide more flexibility, though,
primarily because you can pass them input values
and receive output values.
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Note of Live Functions
• Create Live Function from Selected Code
• If you have an existing large live script or function, you can
break it into smaller pieces by automatically converting
selected areas of code into functions or local functions. This
is called code refactoring.
• To refactor a selected area of code, select one or more lines
of code and on the Live Editor tab, in the Code section, click
Refactor. Then, select from the available options. MATLAB
creates a function with the selected code and replaces the
original code with a call to the newly created function.
• You can run live functions using several methods, including
calling them from the Command Window or calling them
from a live script.
• Disadvantage: can’t run function directly and thus won’t
visualize it results to the right of the code as in live scirpts.
❖Examples: addmefct.m, addme.mlx, RefactorBaseForExp.mlx
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Navigate a live script with shortcuts
Ctrl + Alt + Enter
F5
F10
Shift + F5
Ctrl + Enter
Ctrl + Shift + Enter
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Linear Algebra
• Linear algebra functions in MATLAB® provide
fast, numerically robust matrix calculations.
Capabilities include a variety of:
– matrix operations
– linear equation solving,
– matrix factorizations,
– computation of eigenvalues or singular values,
and more.
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Matrices in the MATLAB Environment
• The MATLAB environment uses the term
matrix to indicate a variable containing real or
complex numbers arranged in a twodimensional grid.
• An array is, more generally, a vector, matrix, or
higher dimensional grid of numbers. All arrays
in MATLAB are rectangular, in the sense that
the component vectors along any dimension
are all the same length.
• The mathematical operations defined on
matrices are the subject of linear algebra.
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Matrices in the MATLAB Environment
• MATLAB has many functions that create
different kinds of matrices, recall zeros, ones
• M = magic(n) returns an n-by-n matrix
constructed from the integers 1 through n2
with equal row and column sums. The order n
must be a scalar greater than or equal to 3 in
order to create a valid magic square.
• X = randi(imax,sz1,sz2) returns an sz1-by-sz2
matrix of pseudorandom integers between
between 1 and imax.
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Matrices in the MATLAB Environment
• I = eye(n,m) returns an n-by-m matrix with
ones on the main diagonal and zeros
elsewhere: identify matrix.
• D = diag(v) returns a square diagonal matrix
with the elements of vector v on the main
diagonal.
• P = pascal(n) returns a Pascal’s Matrix of order
n. P is a symmetric positive definite matrix
with integer entries taken from Pascal's
triangle. The inverse of P has integer entries.
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Matrices in the MATLAB Environment
• Pascal’s triangle is a triangle formed by rows
of numbers. The first row has entry 1. Each
succeeding row is formed by adding adjacent
entries of the previous row, substituting a 0
where no adjacent entry exists. The pascal
function forms Pascal’s matrix by selecting the
portion of Pascal’s triangle that corresponds to
the specified matrix dimensions, as outlined in
the graphic. The matrix outlined corresponds
to the MATLAB® command pascal(4).
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Matrices in the MATLAB Environment
• The matrix outlined corresponds to the
MATLAB® command pascal(4).
❖Example: MatrixCreateExp.mlx
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Matrix Operations
• Matrix operations follow the rules of linear algebra and
are not compatible with multidimensional arrays.
• The required size and shape of the inputs in relation to
one another depends on the operation.
• For nonscalar inputs, the matrix operators generally
calculate different answers than their array operator
counterparts.
• For example, if you use the matrix right division
operator, /, to divide two matrices, the matrices must
have the same number of columns. But if you use the
matrix multiplication operator, *, to multiply two
matrices, then the matrices must have a common inner
dimension. That is, the number of columns in the first
input must be equal to the number of rows in the
second input.
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Matrix Operations
• The matrix multiplication operator calculates
the product of two matrices with the formula:
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Matrix Operations
• The following table provides a summary of matrix
arithmetic operators in MATLAB. For function-specific
information, click the link to the function reference page in
the last column.
Operator
Purpose
Description
Reference Page
*
Matrix multiplication C = A*B is the linear algebraic product of the
matrices A and B. The number of columns of A must
equal the number of rows of B.
\
Matrix left division
x = A\B is the solution to the equation Ax = B.
mldivide
Matrices A and B must have the same number of rows.
/
Matrix right division
x = B/A is the solution to the equation xA = B.
Matrices A and B must have the same number of
columns. In terms of the left division operator, B/A =
(A'\B')'.
^
Matrix power
A^B is A to the power B, if B is a scalar. For other values mpower
of B, the calculation involves eigenvalues and
eigenvectors.
'
Complex conjugate
transpose
A' is the linear algebraic transpose of A. For complex
matrices, this is the complex conjugate transpose.
mtimes
mrdivide
ctranspose
❖Example: MatrixOpExp.mlx
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Matrix Operations
• L = tril(A) returns the lower triangular portion
of matrix A.
• L = tril(A,k) returns the elements on and below
the kth diagonal of A.
• U = triu(A) returns the upper triangular
portion of matrix A.
• U = triu(A,k) returns the elements on and
above the kth diagonal of A.
❖Example: MatrixOpExp.mlx
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References
• https://www.mathworks.com/help/matlab/line
ar-algebra.html
• https://www.mathworks.com/help/matlab/
• Introduction to MATLAB, Ossman K., Bucks G.
2014.
• An Engineering Guide to MATLAB® with
Applications from Mechanical, Aerospace,
Electrical, Civil and Biological System
Engineering, Magrab E., Azarm S., Balachandran
B., Duncan J., Herold K., Walsh G., 3rd ed.
Prentice Hall, 2011.
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