The set ℳm,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:
Since the set of all matrices admits the operation of multiplication in addition to the basic operation of addition (which is included in the definition of vector spaces), it is natural to require that matrix norm satisfies the special property:
Once a norm is defined, it is the most natural way of measure distance between two matrices A and B as d(A, B) = ‖A − B‖ = ‖B − A‖. However, not all distance functions have a corresponding norm. For example, a trivial distance that has no equivalent norm is d(A, A) = 0 and d(A, B) = 1 if A ≠ B.
The norm of a matrix may be thought of as its magnitude or length because it is a nonnegative number.
Their definitions are summarized below for an \( m \times n \) matrix A, to which corresponds a self-adjoint (m+n)×(m+n) matrix B:
This matrix norm is called the operator norm or induced norm.
The term "induced" refers to the fact that the definition of a norm for vectors such as A x and x is what enables the definition above of a matrix norm.
This definition of matrix norm is not computationally friendly, so we use other options. The most important norms are as follow
✼
The operator norm corresponding to the p-norm for vectors, p ≥ 1, is:
Theorem 5:
Let ‖ ‖ be any matrix norm, and let matrix
I + B is singular, where
I is the identity matrix.
Then ‖B‖ ≥ 1 for every matrix norm.
Mathematica 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, Infinity] for evaluating the ∞-norm; Norm[A, "Frobenius"] for evaluating the Frobenius norm.
A = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}
Norm[A]
Sqrt[3/2 (95 + Sqrt[8881])]
N[%]
16.8481
Example 3:
Evaluate the norms of the matrix
\( {\bf A} = \left[ \begin{array}{cc} \phantom{-}1 & -7 & 4 \\ -2 & -3 & 1\end{array} \right] . \)
The absolute column sums of A are \( 1 + | -2 | =3 \) , \( |-7| + | -3 | =10 , \) and \( 4+1 =5 . \)
The larger of these is 10 and therefore \( \| {\bf A} \|_1 = 10 . \)
Norm[A, 1]
10
The absolute row sums of A are \( 1 + | -7 | + 4 =12 \) and
\( | -2 | + |-3| + 1 = 6 ; \) therefore, \( \| {\bf A} \|_{\infty} = 12 . \)
Norm[Transpose[A], 1]
12
The Euclidean norm of A
is the largest singular value. So we calculate
This matrix\( {\bf A}\,{\bf A}^{\ast} \) has two eigenvalues
\( 40 \pm \sqrt{1205} . \) Hence, the Euclidean norm of the matrix A is
\( \sqrt{40 + \sqrt{1205}} \approx 8.64367 . \)
These matrices S and M have the same eigenvalues. Therefore, we found the Euclidean (operator) norm of A to be approximately 16.8481. Mathematica knows this norm:
Norm[A]
Sqrt[3/2 (95 + Sqrt[8881])]
The spectral radius of A is the largest eigenvalue: