# The Matrix Exponent of the Identity Matrix, I

• mhsd91
In summary, the conversation discusses the Matrix Exponent of the Identity Matrix, I, and how it relates to the exponential of a matrix. The correct formula for the Matrix Exponent is e^{aI} = I \left( \sum_{k=0}^{\infty} \frac{a^k}{k!} \right) = I e^{a}, while a suggested solution claims that e^{aI} = e^{-a} I. The conversation ends with the person asking for clarification and questioning if it could be a misprint in the solution.
mhsd91
So, essentially, all I wonder is: What is the The Matrix Exponent of the Identity Matrix, $I$?

Silly question perhaps, but here follows my problem. Per definition, the Matrix Exponent of the matrix $A$ is,

$e^{A} = I + A + \frac{A^2}{2} + \ldots = I + \sum_{k=1}^{\infty} \frac{A^k}{k!} = \sum_{k=0}^{\infty} \frac{A^k}{k!}$

as $e^0 = I$. I suspected that, since $I^k = I$ for any integer $k$, we would get

$e^{I} = I + I + \frac{I}{2} + \ldots = I \cdot \left( \sum_{k=0}^{\infty} \frac{1}{k!} \right) = I \cdot e,\quad e\approx 2.72$

such that for an arbitrary constant $a$ we could write

$e^{aI} = I \left( \sum_{k=0}^{\infty} \frac{a^k}{k!} \right) = I e^{a}$

However, apparently this is not the case as a (suggested) solution to some (homework) problem I've been working on claims that

$e^{aI} = e^{-a} I$

With a sign change of a! I think I'm just missing something trivial and fundamental, but I'd really appreciate some help to sort this one out. Might it also be a misprint in the solution?

mhsd91 said:
$e^{aI} = I \left( \sum_{k=0}^{\infty} \frac{a^k}{k!} \right) = I e^{a}$
This is correct.

However, apparently this is not the case as a (suggested) solution to some (homework) problem I've been working on claims that

$e^{aI} = e^{-a} I$

This is wrong.

mhsd91
mhsd91 said:
So, essentially, all I wonder is: What is the The Matrix Exponent of the Identity Matrix, $I$?

Silly question perhaps, but here follows my problem. Per definition, the Matrix Exponent of the matrix $A$ is,

$e^{A} = I + A + \frac{A^2}{2} + \ldots = I + \sum_{k=1}^{\infty} \frac{A^k}{k!} = \sum_{k=0}^{\infty} \frac{A^k}{k!}$

as $e^0 = I$. I suspected that, since $I^k = I$ for any integer $k$, we would get

$e^{I} = I + I + \frac{I}{2} + \ldots = I \cdot \left( \sum_{k=0}^{\infty} \frac{1}{k!} \right) = I \cdot e,\quad e\approx 2.72$

such that for an arbitrary constant $a$ we could write

$e^{aI} = I \left( \sum_{k=0}^{\infty} \frac{a^k}{k!} \right) = I e^{a}$

However, apparently this is not the case as a (suggested) solution to some (homework) problem I've been working on claims that

$e^{aI} = e^{-a} I$

With a sign change of a! I think I'm just missing something trivial and fundamental, but I'd really appreciate some help to sort this one out. Might it also be a misprint in the solution?
@mhsd91, when you post a question, please do not delete the three parts of the homework template. The template is required.

mhsd91

## What is the Matrix Exponent of the Identity Matrix, I?

The Matrix Exponent of the Identity Matrix, I, is a mathematical concept that involves raising the identity matrix to a given power. The identity matrix, denoted as I, is a special square matrix with 1s on the main diagonal and 0s everywhere else.

## How is the Matrix Exponent of the Identity Matrix, I, calculated?

The Matrix Exponent of the Identity Matrix, I, is calculated by raising each element of the identity matrix to the given power. For example, if the power is 2, then each element of the identity matrix will be squared.

## What is the significance of the Matrix Exponent of the Identity Matrix, I?

The Matrix Exponent of the Identity Matrix, I, has many applications in mathematics and physics. It is used to solve systems of linear equations, compute transformations, and analyze Markov chains.

## Can the Matrix Exponent of the Identity Matrix, I, be negative?

Yes, the Matrix Exponent of the Identity Matrix, I, can be negative. This means that the inverse of the identity matrix is being raised to the given power. In this case, the resulting matrix will have the same size as the identity matrix, but the elements will be the reciprocals of the original elements.

## What is the Matrix Exponent of the Identity Matrix, I, used for in computer science?

In computer science, the Matrix Exponent of the Identity Matrix, I, is used for various tasks such as image processing, data compression, and solving optimization problems. It is also used in machine learning algorithms and artificial intelligence.

Replies
3
Views
783
Replies
6
Views
782
Replies
3
Views
939
Replies
2
Views
1K
Replies
1
Views
771
Replies
14
Views
2K
Replies
3
Views
790
Replies
5
Views
782
Replies
1
Views
639
Replies
5
Views
741