= HESS(A)
produces a unitary matrix P and a Hessenberg matrix H so
that A = P*H*P'. By itself, HESS(A) returns H.
HILB Inverse Hilbert matrix. HILB(N) is the inverse of the N
by N matrix with elements 1/(i+j-1), which is a famous
example of a badly conditioned matrix. The result is exact
for N less than about 15, depending upon the computer.
IF Conditionally execute statements. Simple form...
IF expression rop expression, statements
where rop is =, <, >, <=, >=, or <> (not equal) . The
statements are executed once if the indicated comparison
between the real parts of the first components of the two
expressions is true, otherwise the statements are skipped.
Example.
IF ABS(I-J) = 1, A(I,J) = -1;
More complicated forms use END in the same way it is used
with FOR and WHILE and use ELSE as an abbreviation for END,
IF expression not rop expression . Example
FOR I = 1:N, FOR J = 1:N, ...
IF I = J, A(I,J) = 2; ELSE IF ABS(I-J) = 1, A(I,J) = -1; ...
ELSE A(I,J) = 0;
MATLAB, page 54
An easier way to accomplish the same thing is
A = 2*EYE(N);
FOR I = 1:N-1, A(I,I+1) = -1; A(I+1,I) = -1;
IMAG IMAG(X) is the imaginary part of X .
INV INV(X) is the inverse of the square matrix X . A warning
message is printed if X is badly scaled or nearly
singular.
KRON KRON(X,Y) is the Kronecker tensor product of X and Y . It
is also denoted by X .*. Y . The result is a large matrix
formed by taking all possible products between the elements
of X and those of Y . For example, if X is 2 by 3, then
X .*. Y is
< x(1,1)*Y x(1,2)*Y x(1,3)*Y
x(2,1)*Y x(2,2)*Y x(2,3)*Y >
The five-point discrete Laplacian for an n-by-n grid can be
generated by
T = diag(ones(n-1,1),1); T = T + T'; I = EYE(T);
A = T.*.I + I.*.T - 4*EYE;
Just in case they might be useful, MATLAB includes
constructions called Kronecker tensor quotients, denoted by
X ./. Y and X .\. Y . They are obtained by replacing the
elementwise multiplications in X .*. Y with divisions.
LINES An internal count is kept of the number of lines of output
since the last input. Whenever this count approaches a
limit, the user is asked whether or not to suppress
printing until the next input. Initially the limit is 25.
LINES(N) resets the limit to N .
LOAD LOAD('file') retrieves all the variables from the file .
See FILE and SAVE for more details. To prepare your own
file for LOADing, change the READs to WRITEs in the code
given under SAVE.
LOG LOG(X) is the natural logarithm of X . See FUN .
Complex results are produced if X is not positive, or has
nonpositive eigenvalues.
LONG Determine output format. All computations are done in
complex arithmetic and double precision if it is available.
SHORT and LONG merely switch between different output
formats.
SHORT Scaled fixed point format with about 5 digits.
LONG Scaled fixed point format with about 15 digits.
SHORT E Floating point format with about 5 digits.
LONG E Floating point format with about 15 digits.
MATLAB, page 55
LONG Z System dependent format, often hexadecimal.
LU Factors from Gaussian elimination. = QR(X) produces an upper triangular matrix R of
the same dimension as X and a unitary matrix Q so that
X = Q*R .

= QR(X) produces a permutation matrix E , an
upper triangular R with decreasing diagonal elements and
a unitary Q so that X*E = Q*R .
By itself, QR(X) returns the output of CQRDC . TRIU(QR(X))
is R .
RAND Random numbers and matrices. RAND(N) is an N by N matrix
with random entries. RAND(M,N) is an M by N matrix with
random entries. RAND(A) is the same size as A . RAND
with no arguments is a scalar whose value changes each time
it is referenced.
Ordinarily, random numbers are uniformly distributed in
the interval (0.0,1.0) . RAND('NORMAL') switches to a
normal distribution with mean 0.0 and variance 1.0 .
RAND('UNIFORM') switches back to the uniform distribution.
MATLAB, page 57
RAND('SEED') returns the current value of the seed for the
generator. RAND('SEED',n) sets the seed to n .
RAND('SEED',0) resets the seed to 0, its value when MATLAB
is first entered.
RANK Rank. K = RANK(X) is the number of singular values of X
that are larger than NORM(SIZE(X),'inf')*NORM(X)*EPS.
K = RANK(X,tol) is the number of singular values of X that
are larger than tol .
RCOND RCOND(X) is an estimate for the reciprocal of the
condition of X in the 1-norm obtained by the LINPACK
condition estimator. If X is well conditioned, RCOND(X)
is near 1.0 . If X is badly conditioned, RCOND(X) is
near 0.0 .

__ = SCHUR(X) produces an upper
triangular matrix T , with the eigenvalues of X on the
diagonal, and a unitary matrix U so that X = U*T*U' and
U'*U = EYE . By itself, SCHUR(X) returns T .
SHORT See LONG .
SEMI Semicolons at the end of lines will cause, rather than
suppress, printing. A second SEMI restores the initial
interpretation.
SIN SIN(X) is the sine of X . See FUN .
MATLAB, page 59
SIZE If X is an M by N matrix, then SIZE(X) is .
Can also be used with a multiple assignment,
= SIZE(X) .
SQRT SQRT(X) is the square root of X . See FUN . Complex
results are produced if X is not positive, or has
nonpositive eigenvalues.
STOP Use EXIT instead.
SUM SUM(X) is the sum of all the elements of X .
SUM(DIAG(X)) is the trace of X .
SVD Singular value decomposition. __