This help page is divided into three different types of routines:
| Function | Comments |
|---|---|
| size | returns a row vector containing the row and column dimensions of a matrix |
| length | returns the maximum of the row and column dimensions, useful for vectors |
The function size(A) returns the row and column dimensions of the matrix A as a row vector. A simple way of remembering that the row dimension comes first is to remember the expression down the stairs and into the crypt, that is, the rows come first (down) followed by the columns (across).
To return just the column or row dimension, use a second argument of either 1 or 2, respectively.
>> A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
>> size( A )
ans =
2 3
>> size( A, 1 )
ans =
2
In order to find the number of elements in the matrix, just call
prod( size( A ) ):
>> prod( size( A ) )
ans =
6
length
The length function returns the maximum of the column or row
dimensions. It is useful for finding the dimension of a vector
which may be either an n × 1 column vector, or an
1 × n row vector. For a matrix A, it
is equivalent to max(size(A)).
>> v = [1 2 3 4 5]
v =
1 2 3 4 5
>> length( v )
ans =
5
Sorts, Sums, and Products
| Function | Comments |
|---|---|
| sum | the sum of the entries of a matrix |
| cumsum | the cummulative sums of the entries of a vector |
| prod | the sum of the entries of a matrix |
| cumprod | the cummulative products of the entries of a vector |
| sort | sorts vector or matrix (matrices by columns) |
| sortrows | sorts a matrix according by rows |
| cplxpair | sorts reals and complex conjugate pairs by real part |
The sum function sums the entries of either a row or column vector. For a matrix, it will sum the columns of the matrix, producing a row vector. In order to sum the entries of a matrix A, you may use sum( sum( A ) ). To specifically sum along columns or rows, you may specify a second argument 1 or 2, respectively.
>> sum( [1 2 3 4 5] )
ans =
15
>> A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
>> sum( A )
ans =
5 7 9
>> sum( A, 2 )
ans =
6
15
>> sum( sum( A ) )
ans =
21
The cumsum function returns the cummulative sums of the entries of either a row or column vector. For a matrix, it returns the cummulative sums along the columns of the matrix. To specify taking the cummulative sums along columns or rows, use a second argument 1 or 2, respectively.
The size of the resulting matrix is equal to the size of the input matrix.
>> cumsum( [1 2 3 4 5] )
ans =
1 3 6 10 15
>> A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
>> cumsum( A )
ans =
1 2 3
5 7 9
>> cumsum( A, 2 )
ans =
1 3 6
4 9 15
The prod function sums the entries of either a row or column vector. For a matrix, it will take the product of the columns of the matrix, producing a row vector. In order to take the product of the entries of a matrix A, you may use prod( prod( A ) ). To specifically take the product along columns or rows, you may specify a second argument 1 or 2, respectively.
>> prod( [1 2 3 4 5] ) % 5!
ans =
120
>> A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
>> prod( A )
ans =
4 10 18
>> prod( A, 2 )
ans =
6
120
>> prod( prod( A ) )
ans =
720
The cumprod function returns the cummulative products of the entries of either a row or column vector. For a matrix, it returns the cummulative products along the columns of the matrix. To specify taking the cummulative products along columns or rows, use a second argument 1 or 2, respectively.
The size of the resulting matrix is equal to the size of the input matrix.
>> cumprod( [1 2 3 4 5] )
ans =
1 2 6 24 120
>> A = [1 2 3; 4 5 6]
A =
1 2 3
4 5 6
>> cumprod( A )
ans =
1 2 3
4 10 18
>> cumprod( A, 2 )
ans =
1 2 6
4 20 120
For vectors, sort sorts the entries from smallest to largest. For a matrix, each column is sorted separately.
Complex values are sorted by real part first, then by magnitude of the imaginary part. Thus [1 2 2-i 2+2i 3] is sorted.
To sort in reverse order, do a sort, and then use fliplr or flipud.
To sort the rows of a matrix A, use sort( A' )'.
>> sort( [1 5 3 4] )
ans =
1 3 4 5
>> A = [1 3 2; 2 1 3]
A =
1 3 2
2 1 3
>> sort( A )
ans =
1 1 2
2 3 3
For a matrix or vector, sortrows orders the rows of a matrix so that the first column of the matrix is sorted from smallest to largest. If the first column has equal entries, the entries in subsequent columns are used to break the tie.
In order to sort along a particular column, use a second integer argument to indicate which column to sort along.
>> A = [2 3 5; 1 5 6; 2 4 7; 3 5 8; 2 1 9]
A =
2 3 5
1 5 6
2 4 7
3 5 8
2 1 9
>> sortrows( A )
ans =
1 5 6
2 1 9
2 3 5
2 4 7
3 5 8
>> sortrows( A, 2 )
ans =
2 1 9
2 3 5
2 4 7
1 5 6
3 5 8
The function cplxpair sorts a list of real and complex conjugate pairs (numbers of the form a + bi and a - bi) into purely real numbers and complex conjugate pairs. The complex conjugate pairs are then sorted by their real part and the size of the imaginary part, and these are followed by all real numbers sorted from smallest to largest.
This function is usually used to sort the roots of a real valued polynomial or the eigenvalues of a real matrix.
>> eig( rand( 6 ) ) ans = 3.1830 -0.0609 + 0.3121i -0.0609 - 0.3121i 0.1361 0.7433 + 0.1790i 0.7433 - 0.1790i >> cplxpair( ans ) ans = -0.0609 - 0.3121i -0.0609 + 0.3121i 0.7433 - 0.1790i 0.7433 + 0.1790i 0.1361 3.1830
| Function | Comments |
|---|---|
| dot | the dot product of two vectors |
| cross | the cross product of two vectors |
| norm | the norm of a vector or a square matrix |
| cond | the condition number of a matrix |
| eig | the eigenvalues and eigenvectors of a square matrix |
| inv | the inverse of a square matrix |
| rank | the rank of a matrix |
| det | the determinant of a square matrix |
| trace | the sum of the diagonal elements of a matrix |
| svd | the singular value decomposition of a matrix |
| orth | an orthonormal basis for the range of a matrix |
The dot function takes the dot product of two vectors. If the vectors u and v are both column or both row vectors, then dot(u, v) is equivalent to u'*v and u*v', respectively.
>> u = [1 3 5], v = [5 2 3]
u =
1 3 5
v =
5 2 3
>> dot( u, v )
ans =
26
>> u*v'
ans =
26
The dot function takes the cross product of two vectors. The orientation (row or column) is taken from the first argument.
>> u = [1 3 5], v = [5 2 3]
u =
1 3 5
v =
5 2 3
>> cross( u, v )
ans =
-1 22 -13
The norm(v) function takes Euclidean norm of the vector v. For a matrix A, the norm nA = norm( A ) is the smallest value such that norm(A*v) <= nA*norm(v) for all possible vectors.
The Euclidean norm is also known as the 2-norm. For information about other norms, see help norm.
>> norm( [1 2 3 5] )
ans =
6.2450
>> norm( [1 2 3; 5 3 5; 3 2 5] )
ans =
10.3929
The function cond(A) returns the condition number of the matrix A, which is by definition norm(A) * norm(inv(A)). A large condition number indicate the matrix is nearly singular.
>> cond( [1 2 3; 5 3 5; 3 2 5] ) ans = 14.0635
The function eig(A) returns the eigenvalues of the matrix A as a column vector. If the output is assigned to a row vector of two variables, the first variable is assigned a matrix, the columns of which are the eigenvectors of A, and the second variable a diagonal matrix, the entries of which are the corresponding eigenvalues.
>> A = [1 2 3; 5 3 5; 3 2 5]
A =
1 2 3
5 3 5
3 2 5
>> eig( A )
ans =
9.4987
-1.4006
0.9020
>> [EVec, EVal] = eig( A )
EVec =
-0.3740 -0.7241 -0.2071
-0.7289 0.6777 -0.8109
-0.5734 0.1276 0.5473
EVal =
9.4987 0 0
0 -1.4006 0
0 0 0.9020
Thus, the eigenvalue, eigenvector pairs are:
| 9.4987 | [-0.3740 -0.7289 -0.5734] |
| -1.4006 | [-0.7241 0.6777 0.1276] |
| 0.9020 | [-0.2071 -0.8109 0.5473] |
The function inv( A ) takes the inverse of the matrix A if the inverse exists.
>> A = [1 2 5; 3 2 5; 3 2 2]
A =
1 2 5
3 2 5
3 2 2
>> inv( A )
ans =
-0.5000 0.5000 0.0000
0.7500 -1.0833 0.8333
0 0.3333 -0.3333
>> ans * A
ans =
1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
0 0 1.0000
The rank of a matrix A is the number of linearly independent rows or columns. The function rank( A ) estimates this value.
>> A = [1 4 1; 2 5 3; 4 7 7; 2 2 4]
A =
1 4 1
2 5 3
4 7 7
2 2 4
>> rank( A )
ans =
2
The function det( A ) returns the determinant of the square matrix A. The determinant is the change in volume which occurs under the transformation.
The determinant is the product of the eigenvalues.
>> A = [1 2 3; 5 3 5; 3 2 5]
A =
1 2 3
5 3 5
3 2 5
>> det( A )
ans =
-12
>> prod( eig( A ) )
ans =
-12.0000
The function trace( A ) returns the sum of the diagonal entries of the matrix A. It is equivalent to sum( diag( A ) ).
The trace is the sum of the eigenvalues.
>> A = [1 2 3; 5 3 5; 3 2 5]
A =
1 2 3
5 3 5
3 2 5
>> trace( A )
ans =
9
>> sum( eig( A ) )
ans =
9.0000
The function svd( A ) of a matrix A returns the singular values of the matrix A. If the output is assigned to a row vector of three variables, the variables are assigned matrices U, S, and V where U and V are unitary and S is a matrix the diagonal entries of which are the singular values.
>> A = [1 2 3; 5 3 5; 3 2 5]
A =
1 2 3
5 3 5
3 2 5
>> svd( A )
ans =
10.3929
1.5624
0.7390
>> [U, S, V] = svd( A )
U =
-0.3405 0.7239 -0.6000
-0.7331 -0.6040 -0.3126
-0.5887 0.3335 0.7364
S =
10.3929 0 0
0 1.5624 0
0 0 0.7390
V =
-0.5554 -0.8292 0.0624
-0.3904 0.1938 -0.9000
-0.7342 0.5242 0.4314
>> U*S*V'
ans =
1.0000 2.0000 3.0000
5.0000 3.0000 5.0000
3.0000 2.0000 5.0000
The function orth( A ) returns a matrix, the columns of which are an orthonormal basis for the range of a matrix A.
The rank of the matrix A is the column dimension of the resulting matrix.
>> A = [1 3 5; 2 4 4; 0 1 3]
A =
1 3 5
2 4 4
0 1 3
>> orth( A )
ans =
-0.6667 0.3333
-0.6667 -0.6667
-0.3333 0.6667
>> rank( A )
ans =
2