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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.

Symbolic Math Toolbox™ software extends the capabilities of MATLAB software to matrices of mathematical expressions.

MATLAB has dozens of functions that create different kinds of matrices. There are two functions you can use to create a pair of 3-by-3 example matrices for use throughout this chapter. The first example is symmetric:

A = pascal(3) A = 1 1 1 1 2 3 1 3 6

The second example is not symmetric:

B = magic(3) B = 8 1 6 3 5 7 4 9 2

Another example is a 3-by-2 rectangular matrix of random integers:

C = fix(10*rand(3,2)) C = 9 4 2 8 6 7

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. The statements

u = [3; 1; 4] v = [2 0 -1] s = 7

produce a column vector, a row vector, and a scalar:

u = 3 1 4 v = 2 0 -1 s = 7

For more information about creating and working with matrices, see Creating and Concatenating Matrices.

Addition and subtraction of matrices is defined just as it is
for arrays, element by element. Adding `A`

to `B`

,
and then subtracting `A`

from the result recovers `B`

:

A = pascal(3); B = magic(3); X = A + B X = 9 2 7 4 7 10 5 12 8 Y = X - A Y = 8 1 6 3 5 7 4 9 2

Addition and subtraction require both matrices to have the same dimension, or one of them be a scalar. If the dimensions are incompatible, an error results:

C = fix(10*rand(3,2)) X = A + C Error using + Matrix dimensions must agree.

A row vector and a column vector of the same length can be multiplied
in either order. The result is either a scalar, the *inner* product,
or a matrix, the *outer product* :

u = [3; 1; 4]; v = [2 0 -1]; x = v*u x = 2 X = u*v X = 6 0 -3 2 0 -1 8 0 -4

For real matrices, the *transpose* operation
interchanges *a*_{ij} and *a*_{ji}. MATLAB uses
the apostrophe operator (`'`

) to perform a complex
conjugate transpose, and uses 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`

is *symmetric*,
so `A'`

is equal to `A`

. But, `B`

is
not symmetric:

B = magic(3); X = B' X = 8 3 4 1 5 9 6 7 2

Transposition turns a row vector into a column vector:

x = v' x = 2 0 -1

If `x`

and `y`

are both real
column vectors, the product `x*y`

is not defined,
but the two products

x'*y

and

y'*x

are the same scalar. This quantity is used so frequently, it
has three different names: *inner* product, *scalar* product,
or *dot* product.

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. So if

z = [1+2i 7-3i 3+4i; 6-2i 9i 4+7i] z = 1.0000 + 2.0000i 7.0000 - 3.0000i 3.0000 + 4.0000i 6.0000 - 2.0000i 0 + 9.0000i 4.0000 + 7.0000i

then

z' ans = 1.0000 - 2.0000i 6.0000 + 2.0000i 7.0000 + 3.0000i 0 - 9.0000i 3.0000 - 4.0000i 4.0000 - 7.0000i

The unconjugated complex transpose, where the complex part of
each element retains its sign, is denoted by `z.'`

:

z.' ans = 1.0000 + 2.0000i 6.0000 - 2.0000i 7.0000 - 3.0000i 0 + 9.0000i 3.0000 + 4.0000i 4.0000 + 7.0000i

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.

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:

A = pascal(3); B = magic(3); m = 3; n = 3; for i = 1:m for j = 1:n C(i,j) = A(i,:)*B(:,j); end end

MATLAB uses a single asterisk to denote matrix multiplication.
The next two examples illustrate the fact that matrix multiplication
is not commutative; *AB* is usually not equal to *BA*:

X = A*B X = 15 15 15 26 38 26 41 70 39 Y = B*A Y = 15 28 47 15 34 60 15 28 43

A matrix can be multiplied on the right by a column vector and on the left by a row vector:

u = [3; 1; 4]; x = A*u x = 8 17 30 v = [2 0 -1]; y = v*B y = 12 -7 10

Rectangular matrix multiplications must satisfy the dimension compatibility conditions:

C = fix(10*rand(3,2)); X = A*C X = 17 19 31 41 51 70 Y = C*A Error using * Inner matrix dimensions must agree.

Anything can be multiplied by a scalar:

s = 7; w = s*v w = 14 0 -7

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 *A**I* = *A* and *I**A* = *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

eye(m,n)

returns an *m*-by-*n* rectangular
identity matrix and `eye(n)`

returns an *n*-by-*n* square
identity matrix.

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 in the following order:

[X(1,1)*Y X(1,2)*Y . . . X(1,n)*Y . . . X(m,1)*Y X(m,2)*Y . . . X(m,n)*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 matrix

X = 1 2 3 4

and `I = eye(2,2)`

is the 2-by-2 identity matrix,
then the two matrices

kron(X,I)

and

kron(I,X)

are

1 0 2 0 0 1 0 2 3 0 4 0 0 3 0 4

and

1 2 0 0 3 4 0 0 0 0 1 2 0 0 3 4

The *p*-norm of a vector *x*,

$${\Vert x\Vert}_{p}={\left({\displaystyle \sum {\left|{x}_{i}\right|}^{p}}\right)}^{1/p},$$

is computed by `norm(x,p)`

. This is defined
by 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*:

v = [2 0 -1]; [norm(v,1) norm(v) norm(v,inf)] ans = 3.0000 2.2361 2.0000

The *p*-norm of a matrix *A*,

$${\Vert A\Vert}_{p}=\underset{x}{\mathrm{max}}\frac{{\Vert Ax\Vert}_{p}}{{\Vert x\Vert}_{p}},$$

can be computed for *p* = 1, 2, and ∞
by `norm(A,p)`

. Again, the default value is *p* =
2:

C = fix(10*rand(3,2)); [norm(C,1) norm(C) norm(C,inf)] ans = 19.0000 14.8015 13.0000

MATLAB software 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.

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.