# Upper bound to 2-norm of a matrix

• Simfish
In summary: A||_\infty = \max_i \sum_{j=1}^{n} |a_{ij}|Using the Cauchy-Schwarz inequality, we have:\sum_{i=1}^{m} |a_{ij}| \sum_{j=1}^{n} |a_{ij}| \geq \left(\sum_{i=1}^{m} |a_{ij}| \right)^2Therefore, we have:\max_i \sum_{j=1}^{n} |a_{ij}| \leq \sqrt{\sum_{i=1}^{m} |
Simfish
Gold Member

## Homework Statement

Using the fact that $$||A||_2 = \sqrt { \rho ( A^* A )}$$, prove that
$$||A||_2 \leq \sqrt { ||A||_1 ||A||_\infty }$$. This is an easy estimate to
find in practice for an upper bound on $$||A||_2$$.

## The Attempt at a Solution

Or, in other words, the magnitude of the largest eigenvalue of A*A <= the product of the maximum row sum and the maximum column sum.

So I've been trying to proceed and I'm not totally sure what to do. One thing I've figured: equality exists when the matrix is a diagonal matrix (so the entire matrix's values occur where they contribute to the eigenvalues).

There is an inequality $$||A||_2 < \sqrt{m} * ||A||_\infty$$ but I doubt this would help since the inequality would contain an extra factor of $$\sqrt{m}$$ (and an extra factor of sqrt(n) too, once you accounted for the 1-norm)

it is important to understand the mathematical principles behind the equations and not just rely on practical estimates. Therefore, let us start by understanding what the 2-norm, 1-norm, and infinity-norm represent.

The 2-norm of a matrix A, denoted by ||A||_2, is the maximum singular value of A. This is equivalent to the square root of the largest eigenvalue of A^*A, where A^* is the conjugate transpose of A. In other words, it represents the maximum stretch factor of A in any direction.

The 1-norm of a matrix A, denoted by ||A||_1, is the maximum absolute column sum of A. This represents the maximum absolute value of the sum of the elements in any column of A.

The infinity-norm of a matrix A, denoted by ||A||_\infty, is the maximum absolute row sum of A. This represents the maximum absolute value of the sum of the elements in any row of A.

Now, let us prove the given inequality using the fact that ||A||_2 = \sqrt { \rho ( A^* A )}. We can rewrite the inequality as:

\sqrt { \rho ( A^* A )} \leq \sqrt { ||A||_1 ||A||_\infty }

Squaring both sides, we get:

\rho ( A^* A ) \leq ||A||_1 ||A||_\infty

Now, let us consider the spectral decomposition of A^*A, which states that A^*A = U \Lambda U^*, where U is a unitary matrix and \Lambda is a diagonal matrix with the eigenvalues of A^*A on the diagonal. Therefore, we have:

\rho ( A^* A ) = \rho ( U \Lambda U^* ) = \rho ( \Lambda )

Using the fact that the maximum eigenvalue of a diagonal matrix is the maximum value on its diagonal, we can rewrite this as:

\rho ( A^* A ) \leq \max_i \lambda_i

Where \lambda_i represents the i-th diagonal element of \Lambda. Now, let us consider the maximum row sum and the maximum column sum of A. Using the definition of the 1-norm and infinity-norm, we have:

||A||_1 = \max

## 1. What is an upper bound to 2-norm of a matrix?

The upper bound to 2-norm of a matrix is a mathematical concept that represents the maximum possible value for the 2-norm of a particular matrix. It is a way to measure the size or magnitude of a matrix, and it can provide important information about the properties and behavior of the matrix.

## 2. How is the upper bound to 2-norm of a matrix calculated?

The upper bound to 2-norm of a matrix is calculated using a mathematical formula that involves finding the square root of the sum of the squares of the matrix's elements. This value is then multiplied by the square root of the matrix's dimension to obtain the upper bound.

## 3. What is the significance of the upper bound to 2-norm of a matrix?

The upper bound to 2-norm of a matrix is significant because it can provide useful information about the matrix's magnitude and its behavior in various mathematical operations. It can also be used to compare the sizes of different matrices and make predictions about their properties.

## 4. Can the upper bound to 2-norm of a matrix be exceeded?

No, the upper bound to 2-norm of a matrix cannot be exceeded. It represents the maximum possible value for the 2-norm of a matrix, and no matrix can have a 2-norm greater than this upper bound.

## 5. How is the upper bound to 2-norm of a matrix used in practical applications?

The upper bound to 2-norm of a matrix is used in various practical applications in fields such as engineering, computer science, and physics. It can help in analyzing data, designing algorithms, and making predictions about the behavior of systems represented by matrices.

Replies
2
Views
870
Replies
12
Views
2K
Replies
9
Views
1K
Replies
1
Views
2K
Replies
2
Views
3K
Replies
1
Views
2K
Replies
9
Views
2K
Replies
4
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K