Linear Algebra: Prove rank(A) <= rank(exp(A))

In summary, the conversation discusses the proof for the inequality of ranks for matrices A and exp(A). The conversation mentions using Sylvester's inequality and the Taylor Series Expansion to prove the inequality. The individual is new to math proofs and is seeking guidance on where to start. They also mention that A should be diagonalizable.
  • #1
aznkid310
109
1

Homework Statement


Let a be all real numbers, nxn. Prove that

a) rank(A) less than or equal to rank(exp(A))
b) rank(exp(A)-I) less than or equal to Rank(A)

Homework Equations



I'm new to math proofs and so don't really know where to start. Could someone point me in the right direction? Would I need to prove that A is diagonalizable and somehow proceed from there?

The Attempt at a Solution



rank(A) + dim N(A) = n, N(A) = nullspasce of A
This means rank(A) less than or equal to n

exp(A) = I + A + (1/2)A^2 +...+ (1/(r-1)!)*A^(r-1) Taylor Series Expansion

Using Sylvester's Inequality: [rank(A) + rank(exp(A)) -n ] less than or equal to rank(Aexp(A))

Aexp(A) = A + A^2 + (1/2)A^3 + ... + (1/(r-1)!)*A^r
 
Last edited:
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  • #2
I don't really have a whole lot of insight right now but thought I'd just point out real quick that you're missing a -n in Sylvester's inequality
 
  • #3
Fixed Thanks.
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, linear transformations, and vector spaces.

What does "rank" refer to in linear algebra?

In linear algebra, the rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix.

What is the significance of proving rank(A) <= rank(exp(A))?

Proving that rank(A) <= rank(exp(A)) is important because it helps us understand the relationship between the ranks of a matrix and its exponential. It also has various applications in fields such as physics, engineering, and computer science.

How do you prove rank(A) <= rank(exp(A))?

To prove rank(A) <= rank(exp(A)), we can use the concept of eigenvalues and eigenvectors. By using the fact that exp(A) has the same eigenvalues as A, we can show that the rank of exp(A) is at least equal to the rank of A.

What are some real-world examples where "Linear Algebra: Prove rank(A) <= rank(exp(A))" is applied?

This concept is applied in various fields such as data analysis, signal processing, and control systems. For example, in data analysis, it can be used to analyze the relationship between two sets of data and determine the strength of the correlation between them.

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