Matrices in the MATLAB Environment
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It is important to determine what is causing this amount of compensation. Check the Fuel Trim 'numbers' over the full range of operating conditions. A healthy engine should have Long Term Fuel Trim numbers around 1 to 3 percent, either positive or negative. Mechanical Misfire. Mechanical problems can also cause an engine to misfire.
This topic contains an introduction to creating matrices and performing basic matrix calculations in MATLAB®.
The MATLAB environment uses the term matrix to indicate a variable containing real or complex numbers arranged in a two-dimensional 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.
Creating Matrices
MATLAB has many functions that create different kinds of matrices. For example, you can create a symmetric matrix with entries based on Pascal's triangle:
Or, you can create an unsymmetric magic square matrix, which has equal row and column sums:
Another example is a 3-by-2 rectangular matrix of random integers. In this case the first input to
randi
describes the range of possible values for the integers, and the second two inputs describe the number of rows and columns.- Check no.1 cylinder's sparkplug/compression/injector Engine Misfiring can be caused by worn or fouled spark plugs, a weak spark (weak coil, bad spark plug wire), loss of compression, vacuum leaks, anything that causes an unusually lean fuel mixture (lean misfire), an EGR valve that is stuck open, dirty fuel injectors, low fuel pressure, or even bad fuel.
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A column vector is an m-by-1 matrix, a row vector is a 1-by-n matrix, and a scalar is a 1-by-1 matrix. To define a matrix manually, use square brackets
[ ]
to denote the beginning and end of the array. Within the brackets, use a semicolon ;
to denote the end of a row. In the case of a scalar (1-by-1 matrix), the brackets are not required. For example, these statements produce a column vector, a row vector, and a scalar:For more information about creating and working with matrices, see Creating, Concatenating, and Expanding Matrices.
Adding and Subtracting Matrices
Addition and subtraction of matrices and arrays is performed element-by-element, or element-wise. For example, adding
A
to B
and then subtracting A
from the result recovers B
:Addition and subtraction require both matrices to have compatible dimensions. If the dimensions are incompatible, an error results:
For more information, see Array vs. Matrix Operations.
Vector Products and Transpose
A row vector and a column vector of the same length can be multiplied in either order. The result is either a scalar, called the inner product, or a matrix, called the outer product:
For real matrices, the transpose operation interchanges aij and aji. For complex matrices, another consideration is whether to take the complex conjugate of complex entries in the array to form the complex conjugate transpose. MATLAB uses the apostrophe operator (
'
) to perform a complex conjugate transpose, and the dot-apostrophe operator (.'
) to transpose without conjugation. For matrices containing all real elements, the two operators return the same result.The example matrix
A = pascal(3)
is symmetric, so A'
is equal to A
. However, B = magic(3)
is not symmetric, so B'
has the elements reflected along the main diagonal:For vectors, transposition turns a row vector into a column vector (and vice-versa):
If
x
and y
are both real column vectors, then the product x*y
is not defined, but the two productsand
produce the same scalar result. This quantity is used so frequently, it has three different names: inner product, scalar product, or dot product. There is even a dedicated function for dot products named
dot
.For a complex vector or matrix,
z
, the quantity z'
not only transposes the vector or matrix, but also converts each complex element to its complex conjugate. That is, the sign of the imaginary part of each complex element changes. For example, consider the complex matrixThe complex conjugate transpose of
z
is:The unconjugated complex transpose, where the complex part of each element retains its sign, is denoted by
z.'
:For complex vectors, the two scalar products
x'*y
and y'*x
are complex conjugates of each other, and the scalar product x'*x
of a complex vector with itself is real.Focus Matrix 1 3 3 Cylinder Head Removal
Multiplying Matrices
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Multiplication of matrices is defined in a way that reflects composition of the underlying linear transformations and allows compact representation of systems of simultaneous linear equations. The matrix product C = AB is defined when the column dimension of A is equal to the row dimension of B, or when one of them is a scalar. If A is m-by-p and B is p-by-n, their product C is m-by-n. The product can actually be defined using MATLAB
for
loops, colon
notation, and vector dot products:MATLAB uses an asterisk to denote matrix multiplication, as in
C = A*B
. Matrix multiplication is not commutative; that is, A*B
is typically not equal to B*A
:A matrix can be multiplied on the right by a column vector and on the left by a row vector:
Rectangular matrix multiplications must satisfy the dimension compatibility conditions. Since
A
is 3-by-3 and C
is 3-by-2, you can multiply them to get a 3-by-2 result (the common inner dimension cancels):However, the multiplication does not work in the reverse order:
You can multiply anything with a scalar:
When you multiply an array by a scalar, the scalar implicitly expands to be the same size as the other input. This is often referred to as scalar expansion.
Identity Matrix
Generally accepted mathematical notation uses the capital letter I to denote identity matrices, matrices of various sizes with ones on the main diagonal and zeros elsewhere. These matrices have the property that AI = A and IA = A whenever the dimensions are compatible.
The original version of MATLAB could not use I for this purpose because it did not distinguish between uppercase and lowercase letters and i already served as a subscript and as the complex unit. So an English language pun was introduced. The function
returns an m-by-n rectangular identity matrix and
eye(n)
returns an n-by-n square identity matrix.Matrix Inverse
If a matrix
A
is square and nonsingular (nonzero determinant), then the equations AX = I and XA = I have the same solution X. This solution is called the inverse of A
and is denoted A-1. The inv
function and the expression A^-1
both compute the matrix inverse.The determinant calculated by
det
is a measure of the scaling factor of the linear transformation described by the matrix. When the determinant is exactly zero, the matrix is singular and no inverse exists.Some matrices are nearly singular, and despite the fact that an inverse matrix exists, the calculation is susceptible to numerical errors. The
cond
function computes the condition number for inversion, which gives an indication of the accuracy of the results from matrix inversion. The condition number ranges from 1
for a numerically stable matrix to Inf
for a singular matrix.It is seldom necessary to form the explicit inverse of a matrix. A frequent misuse of
inv
arises when solving the system of linear equations Ax = b. The best way to solve this equation, from the standpoint of both execution time and numerical accuracy, is to use the matrix backslash operator x = Ab
. See mldivide
for more information.Kronecker Tensor Product
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The Kronecker product,
kron(X,Y)
, of two matrices is the larger matrix formed from all possible products of the elements of X
with those of Y
. If X
is m-by-n and Y
is p-by-q, then kron(X,Y)
is mp-by-nq. The elements are arranged such that each element of X
is multiplied by the entire matrix Y
:The Kronecker product is often used with matrices of zeros and ones to build up repeated copies of small matrices. For example, if
X
is the 2-by-2 matrixand
I = eye(2,2)
is the 2-by-2 identity matrix, then:and
Doubletake 2 6 2 trailer. Aside from
kron
, some other functions that are useful to replicate arrays are repmat
, repelem
, and blkdiag
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Vector and Matrix Norms
The p-norm of a vector x,
is computed by
norm(x,p)
. This operation is defined for any value of p > 1, but the most common values of p are 1, 2, and ∞. The default value is p = 2, which corresponds to Euclidean length or vector magnitude:The p-norm of a matrix A,
can be computed for p = 1, 2, and ∞ by
norm(A,p)
. Again, the default value is p = 2:In cases where you want to calculate the norm of each row or column of a matrix, you can use
vecnorm
:Using Multithreaded Computation with Linear Algebra Functions
MATLAB supports multithreaded computation for a number of linear algebra and element-wise numerical functions. These functions automatically execute on multiple threads. For a function or expression to execute faster on multiple CPUs, a number of conditions must be true:
- The function performs operations that easily partition into sections that execute concurrently. These sections must be able to execute with little communication between processes. They should require few sequential operations.
- The data size is large enough so that any advantages of concurrent execution outweigh the time required to partition the data and manage separate execution threads. For example, most functions speed up only when the array contains several thousand elements or more.
- The operation is not memory-bound; processing time is not dominated by memory access time. As a general rule, complicated functions speed up more than simple functions.
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The matrix multiply
(X*Y)
and matrix power (X^p)
operators show significant increase in speed on large double-precision arrays (on order of 10,000 elements). The matrix analysis functions det
, rcond
, hess
, and expm
also show significant increase in speed on large double-precision arrays.Related Topics
Description of Focus Matrix 1.3.3
Focus Matrix 1.3.3 is a smart personal organizer based on the so-called Eisenhower box, a personal productivity strategy created by Dwight Eisenhower, the 34th President of the United States. The principle uses a special priority matrix that helps break your tasks into several groups depending on their urgency and importance, and work your way through your daily agenda in the most efficient way possible.
The key features:
– Two views: Matrix and List of Sections
– Move tasks between sections using drag-and-drop
– Maximize a section to get a closer look at its content
– Use the “Inbox” section for tasks which are needed to be sorted by quadrants
– Set “All day task” to plan tasks for the whole working day
– Sort tasks by title, due date or manually
– Assign due dates to tasks
– Set reminders to never miss an important deadline
– Tasks that are approaching their due dates are automatically carried over from the Not Urgent category to the Urgent list with user notification
– Delegate tasks by email, messages, etc. (through the context menu)
– Task lists can be shared to Facebook, Twitter, etc.
– Editing of multiple tasks at once
– Copying and pasting of tasks within the program, as well as from/to other applications
– Search and tags support
– Task filtering by tags and statuses (completed/uncompleted)
– Export to PDF or print the task list
– Configurable statistics and reports for completed tasks
– Password protection
Compatibility: OS X 10.9 or later 64 bit
The key features:
– Two views: Matrix and List of Sections
– Move tasks between sections using drag-and-drop
– Maximize a section to get a closer look at its content
– Use the “Inbox” section for tasks which are needed to be sorted by quadrants
– Set “All day task” to plan tasks for the whole working day
– Sort tasks by title, due date or manually
– Assign due dates to tasks
– Set reminders to never miss an important deadline
– Tasks that are approaching their due dates are automatically carried over from the Not Urgent category to the Urgent list with user notification
– Delegate tasks by email, messages, etc. (through the context menu)
– Task lists can be shared to Facebook, Twitter, etc.
– Editing of multiple tasks at once
– Copying and pasting of tasks within the program, as well as from/to other applications
– Search and tags support
– Task filtering by tags and statuses (completed/uncompleted)
– Export to PDF or print the task list
– Configurable statistics and reports for completed tasks
– Password protection
Compatibility: OS X 10.9 or later 64 bit
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Compatibility
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