14.2.2  Special matrices

Identity matrix.

The idn or identity command creates identity matrices.

• idn takes n, a positive integer or
  A, a square matrix.
• idn(n) returns the n× n identity matrix.
• idn(A) returns the identity matrix the same size as A.

Examples

Zero matrix.

The newMat command creates a matrix of all zeros.

• newMat takes two arguments: n and p, two positive integers. newMat(n,p) returns the n× p zero matrix.

Example

Diagonals of matrices and diagonal matrices.

The diag or BlockDiagonal command either creates a diagonal matrix or finds the diagonal elements of an existing matrix or creates diagonal matrices.

• diag takes L, a list or a square matrix.
• diag(L) (for a list L) returns the diagonal matrix with the entries of L on the diagonal.
• diag(L) (for a matrix L) returns a list consisting of the diagonal elements of L.

Examples

Jordan blocks.

The JordanBlock command creates a Jordan block, i.e. a square matrix with the same value for all diagonal elements, ones just above the diagonal, and zeros everywhere else.

• JordanBlock takes two arguments:
  • a, an expression.
  • n, a positive integer.
  JordanBlock(a,n) returns the n× n matrix with as on the principal diagonal, ones above this diagonal and zeros everywhere else.

Example

Hilbert matrix.

A Hilbert matrix is a square matrix whose element in the ith row and jth column (recall that the indices are zero-based) is

The hilbert command creates Hilbert matrices. See Section 21.4.6 for other uses of hilbert.

• hilbert takes n, a positive integer.
• hilbert(n) returns the n× n Hilbert matrix.

Examples

Hilbert matrix is known for being regular but severely ill-conditioned. For example:

Vandermonde matrix.

A Vandermonde matrix is a square matrix where each row starts with a 1 and is in geometric progression. The vandermonde command finds a Vandermonde matrix.

• vandermonde takes X=[x₀,…,x_n−1], a vector.
• vandermonde(X) returns the corresponding Vandermonde matrix; namely, the kth row of the matrix is the vector whose components are x_i^k for i=0,…,n−1 and k=0,…,n−1.

Remark.

Remember that the indices of the rows and columns begin at 0 with Xcas.

Example

If a is symbolic else enter purge(a) before: