Arithmetic operations

Now we start working with matrices. A period “.” can also be used for matrix multiplication between one matrix and a vector or two matrices. It is important to note that when doing matrix multiplication an (m x n) matrix can only be multiplied by an (n x s) where m, n, and s are whole numbers, producing an (m x s) matrix.

matlab knows what vector should be used when a matrix is multiplied by a vector. For example,

Products:

Now we generate a special matrix, called the zero matrix, whose entries are all zeroes.

As a simple introductory example, consider 2×2 matrix of zero trace:

The set ℳ_m,n of all m × n matrices under the field of either real or complex numbers is a vector space of dimension m  · n. In order to determine how close two matrices are, and in order to define the convergence of sequences of matrices, a special concept of matrix norm is employed, with notation \( \| {\bf A} \| . \) A norm is a function from a real or complex vector space to the nonnegative real numbers that satisfies the following conditions:

• Positivity:     ‖A‖ ≥ 0     ‖A‖ = 0 iff A = 0.
• Homogeneity:     ‖kA‖ = |k| ‖A‖ for arbitrary scalar k.
• Triangle inequality:     ‖A + B‖ ≤ ‖A‖ + ‖B‖.

✼ The operator norm corresponding to the p-norm for vectors, p ≥ 1, is:

✼ 1-norm (is commonly known as the maximum column sum norm) of a matrix A may be computed as

✻ The infinity norm, \( \infty - \) norm of matrix A may be computed as

✺ In the special case of p = 2 we get the Euclidean norm (which is equal to the largest singular value of a matrix)

✳ The Frobenius norm (not induced norm):

There is also another function that that provides infinum of all norms of a squar matrix: \( \rho ({\bf A}) \le \|{\bf A}\| . \)

Some properties of the matrix norms are presented in the following

All these norms are equivalent and we present some inequalities:

matlab has a special command for evaluating norms:
norm(A) = norm(A,2) for evaluating the Euclidean norm of the matrix A;
norm(A, 1) for evaluating the 1-norm;
norm(A, inf) for evaluating the ∞-norm;
norm(A, 'fro') for evaluating the Frobenius norm;

• 1-norm
  \[ \| {\bf x} \|_{1} = | x_1 | + |x_2 | + \cdots + |x_n | . \]
• the Chebyshev norm
  \[ \| {\bf x} \|_{\infty} = \max_{1 \le i \le n} \left\{ | x_i | \right\} \]
• the Euclidean norm
  \[ \| {\bf x} \|_{2} = \left( |x_1 |^2 + |x_2 |^2 + \cdots + |x_n |^2 \right)^{1/2} . \]
• p-norm
  \[ \| {\bf x} \|_{p} = \left( |x_1 |^p + |x_2 |^p + \cdots + |x_n |^p \right)^{1/p} . \]

To find the Euclidean length of a vector use the Norm[vector] operation. The command norm( ) also works for finding the norm of a complex number, where the imaginary unit is represented by “I.”

This definition can be extended to matrices:

A = [1 2; 3 4]
B = [5 6; 7 8]
AplusB = A + B
AtimesB = A * B

			AtimeseeB = A .* B

			C=[1 2; 3 4]

			C'

		D=[1+i 2; 3 4]

			D'

			D.'

Theorem: If the sizes of the matrices are such that the stated operations can be performed, then:

• \( \left({\bf A}^T \right)^T = {\bf A} \) for any matrix A;
• \( \left( {\bf A} + {\bf B} \right)^T = {\bf A}^T + {\bf B}^T .\)
• \( \left( {\bf A} - {\bf B} \right)^T = {\bf A}^T - {\bf B}^T .\)
• \( \left( k\,{\bf A} \right)^T = k\, {\bf A}^T .\)
• \( \left( {\bf A} \, {\bf B}\right)^T = {\bf B}^T \, {\bf A}^T . \)

The norm of a matrix may be thought of as its size because it is a nonnegative number. Matrix norms are directly related to vector norms. The definitions are summarized below for an \( m \times n \) matrix A.

1-norm (is commonly known as the maximum column sum norm) of a matrix A may be computed as

Some properties of the matrix norms are presented in the following

matlab has special commands for evaluating norms:
norm(A,1) for evaluating 1-norm of the matrix A;
norm(A,inf) for evaluating \( infty - \)norm of the matrix A;
norm(A,2) for evaluating 2-norm (Euclidean) of the matrix A;

The Frobenius norm:

Complete