Question

Prove: \(\|\mathbf{A B}\| \leq\|\mathbf{A}\|\|\mathbf{B}\|\).

Short answer

Question: Prove the submultiplicative property of the matrix norm: \(\|\mathbf{A B}\| \leq\|\mathbf{A}\|\|\mathbf{B}\|\).

Step-by-step solution

Matrix norm definition

We will be using the definition of a matrix norm: \(\|\mathbf{A}\| \equiv \sup \frac{\|\mathbf{A x}\|}{\|\mathbf{x}\|}\), where \(\sup\) denotes the supremum and the vectors \(\mathbf{x}\) have nonzero length.

Expand the matrix product

We need to expand the matrix product \(\mathbf{A B}\), where \(\mathbf{A}\) and \(\mathbf{B}\) are two matrices. We can write \(\mathbf{A B}\) as \(\mathbf{A}\left(\mathbf{B x}\right)\) which is matrix \(\mathbf{A}\) multiplying a new vector \(\mathbf{B x}\).

Properties of matrix norm and triangle inequality

Now, using the definition of a matrix norm, we can find the matrix norm for \(\mathbf{A}(\mathbf{B x})\): \(\|\mathbf{A}( \mathbf{B x})\| \leq \|\mathbf{A}\| \|\mathbf{B x}\|\). To make further simplifications, we note that: \(\|\mathbf{B x}\| \leq \|\mathbf{B}\|\|\mathbf{x}\|\).

Rearrange the expression

Combining the results from Step 3, we have: \(\|\mathbf{A}(\mathbf{B x})\| \leq \|\mathbf{A}\| \|\mathbf{B x}\| \leq \|\mathbf{A}\| \|\mathbf{B}\|\|\mathbf{x}\|\). Taking the supremum of the left-hand side over all nonzero \(\mathbf{x}\), we get: \(\|\mathbf{A B}\| \leq\|\mathbf{A}\|\|\mathbf{B}\|\). Hence, we have proven the submultiplicative property of the matrix norm.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a new matrix. To multiply matrices, you must ensure that the number of columns in the first matrix matches the number of rows in the second matrix. This requirement allows the multiplication of corresponding elements and their summation.
While multiplying, each element in the resulting matrix is obtained by taking each row from the first matrix and each column from the second matrix. For element \((i,j)\) in the resulting matrix, the calculation involves the sum of the products of elements of the \((i)\)-th row of the first matrix and the \((j)\)-th column of the second.

• The matrix product \(\mathbf{A} \mathbf{B}\) depends on the rows of \mathbf{A}\ and the columns of \mathbf{B}\.
• Matrix multiplication is associative and distributive but not commutative, meaning \(\mathbf{A} \mathbf{B} \, eq \, \mathbf{B} \mathbf{A}\).

This operation enables transformations in vector spaces, where matrices can represent systems of linear equations or transformations such as rotations and scalings.

Submultiplicative Property

The submultiplicative property of matrix norms states that the norm of the product of two matrices is less than or equal to the product of their norms. Mathematically, this is expressed as \(\|\mathbf{A B}\| \leq \|\mathbf{A}\| \|\mathbf{B}\|\).
This property is important because it ensures stability in numerical calculations and bounds the error propagation in computations involving matrices.

• It is a key feature in analyzing the behavior of matrix operations.
• Helps in understanding how errors might amplify when dealing with sequences of matrix operations.

This property stems from the general definition of matrix norms, which relate the magnitude of matrix transformations applied to vectors. When considering practical applications, this property assures that the outcomes of matrix-linked calculations remain bounded, making it crucial in fields like computational physics and numerical analysis.

Supremum

Supremum, often denoted as "sup," is a concept from mathematical analysis. It refers to the smallest upper bound of a set. In simpler terms, it is the least number that is at least as large as every element in a given set.
When we talk about the supremum of norms, such as in the matrix norm definition, it refers to the greatest value that the fraction \(\frac{\|\mathbf{A x}\|}{\|\mathbf{x}\|}\) can achieve, over all nonzero vectors \mathbf{x}\.

• Supremum might not necessarily be within the set but can act as a boundary point.
• In context of matrix norms, it helps in defining the maximum stretching factor a matrix can apply to any vector.

For students, understanding the supremum helps in conceptualizing maximum potential outcomes of transformations, as it provides a way to describe how much a matrix can magnify a vector, albeit in a theoretical sense.