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### 20.1 Basic Matrix Functions

— Loadable Function: aa = balance (a, opt)
— Loadable Function: [dd, aa] = balance (a, opt)
— Loadable Function: [cc, dd, aa, bb] = balance (a, b, opt)

`[dd, aa] = balance (a)` returns `aa = dd \ a * dd`. `aa` is a matrix whose row and column norms are roughly equal in magnitude, and `dd` = `p * d`, where `p` is a permutation matrix and `d` is a diagonal matrix of powers of two. This allows the equilibration to be computed without roundoff. Results of eigenvalue calculation are typically improved by balancing first.

`[cc, dd, aa, bb] = balance (a, b)` returns `aa = cc*a*dd` and `bb = cc*b*dd)`, where `aa` and `bb` have non-zero elements of approximately the same magnitude and `cc` and `dd` are permuted diagonal matrices as in `dd` for the algebraic eigenvalue problem.

The eigenvalue balancing option `opt` is selected as follows:

`"N"`, `"n"`
No balancing; arguments copied, transformation(s) set to identity.
`"P"`, `"p"`
Permute argument(s) to isolate eigenvalues where possible.
`"S"`, `"s"`
Scale to improve accuracy of computed eigenvalues.
`"B"`, `"b"`
Permute and scale, in that order. Rows/columns of a (and b) that are isolated by permutation are not scaled. This is the default behavior.

Algebraic eigenvalue balancing uses standard Lapack routines.

Generalized eigenvalue problem balancing uses Ward's algorithm (SIAM Journal on Scientific and Statistical Computing, 1981).

— Function File: cond (a)

Compute the (two-norm) condition number of a matrix. `cond (a)` is defined as `norm (a) * norm (inv (a))`, and is computed via a singular value decomposition.

— Loadable Function: [d, rcond] = det (a)

Compute the determinant of a using Lapack. Return an estimate of the reciprocal condition number if requested.

— Function File: dmult (a, b)

If a is a vector of length `rows (`b`)`, return `diag (`a`) * `b (but computed much more efficiently).

— Function File: dot (x, y, dim)

Computes the dot product of two vectors. If x and y are matrices, calculate the dot-product along the first non-singleton dimension. If the optional argument dim is given, calculate the dot-product along this dimension.

— Loadable Function: lambda = eig (a)
— Loadable Function: [v, lambda] = eig (a)

The eigenvalues (and eigenvectors) of a matrix are computed in a several step process which begins with a Hessenberg decomposition, followed by a Schur decomposition, from which the eigenvalues are apparent. The eigenvectors, when desired, are computed by further manipulations of the Schur decomposition.

— Loadable Function: g = givens (x, y)
— Loadable Function: [c, s] = givens (x, y)

Return a 2 by 2 orthogonal matrix g` = [`c s`; -`s`' `c`]` such that g` [`x`; `y`] = [*; 0]` with x and y scalars.

For example,

```          givens (1, 1)
=>   0.70711   0.70711
-0.70711   0.70711
```

— Loadable Function: [x, rcond] = inv (a)
— Loadable Function: [x, rcond] = inverse (a)

Compute the inverse of the square matrix a. Return an estimate of the reciprocal condition number if requested, otherwise warn of an ill-conditioned matrix if the reciprocal condition number is small.

— Function File: norm (a, p)

Compute the p-norm of the matrix a. If the second argument is missing, `p = 2` is assumed.

If a is a matrix:

p = `1`
1-norm, the largest column sum of the absolute values of a.
p = `2`
Largest singular value of a.
p = `Inf`
Infinity norm, the largest row sum of the absolute values of a.
p = `"fro"`
Frobenius norm of a, `sqrt (sum (diag (`a`' * `a`)))`.

If a is a vector or a scalar:

p = `Inf`
`max (abs (`a`))`.
p = `-Inf`
`min (abs (`a`))`.
other
p-norm of a, `(sum (abs (`a`) .^ `p`)) ^ (1/`p`)`.

— Function File: null (a, tol)

Return an orthonormal basis of the null space of a.

The dimension of the null space is taken as the number of singular values of a not greater than tol. If the argument tol is missing, it is computed as

```          max (size (a)) * max (svd (a)) * eps
```

— Function File: orth (a, tol)

Return an orthonormal basis of the range space of a.

The dimension of the range space is taken as the number of singular values of a greater than tol. If the argument tol is missing, it is computed as

```          max (size (a)) * max (svd (a)) * eps
```

— Loadable Function: pinv (x, tol)

Return the pseudoinverse of x. Singular values less than tol are ignored.

If the second argument is omitted, it is assumed that

```          tol = max (size (x)) * sigma_max (x) * eps,
```

where `sigma_max (`x`)` is the maximal singular value of x.

— Function File: rank (a, tol)

Compute the rank of a, using the singular value decomposition. The rank is taken to be the number of singular values of a that are greater than the specified tolerance tol. If the second argument is omitted, it is taken to be

```          tol = max (size (a)) * sigma(1) * eps;
```

where `eps` is machine precision and `sigma(1)` is the largest singular value of a.

— Function File: trace (a)

Compute the trace of a, `sum (diag (`a`))`.