Invertible Matrices and Rank 1 Matrices: Understanding Linear Transpose

In summary, the conversation is discussing three parts regarding matrix operations and invertibility. The first part involves proving that the transpose of the inverse of an invertible matrix is equal to the inverse of the transpose. The second part discusses proving the existence of nonzero vectors that satisfy a specific equation involving a matrix with rank 1. The third part requires proving that if a matrix multiplied by a vector equals zero, then the vector must also equal zero. The conversation also briefly mentions the properties of matrix multiplication and the concept of an outer product.
  • #1
bananasplit
9
0

Homework Statement


I have an idea on how to part 1, but I have no clue on how to do part 2 and 3.

1.Suppose A is invertible. Check that (A-1)TAT=I and AT(A-1)T=I, and deduce that AT is likewise invertible with inverse (A-1)T.

2. Suppose A is an mxn matrix with rank 1. Prove that there are nonzero vectors u element in Rm and v element in Rn such that A=uvT.

3.Suppose A is an mxn matrix and x is an element of Rn satisfies (ATA)x=0. Prove that AX=0.


Homework Equations



For part one I'm guessing (AB)T=BTAT and (A-1A-1)=I

The Attempt at a Solution


Part 1. I know that some kind of way that it is due to the relationship of (A-1A-1)=I
 
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  • #2
bananasplit said:

Homework Statement


I have an idea on how to part 1, but I have no clue on how to do part 2 and 3.

1.Suppose A is invertible. Check that (A-1)TAT=I and AT(A-1)T=I, and deduce that AT is likewise invertible with inverse (A-1)T.

2. Suppose A is an mxn matrix with rank 1. Prove that there are nonzero vectors u element in Rm and v element in Rn such that A=uvT.

3.Suppose A is an mxn matrix and x is an element of Rn satisfies (ATA)x=0. Prove that AX=0.


Homework Equations



For part one I'm guessing (AB)T=BTAT and (A-1A-1)=I

The Attempt at a Solution


Part 1. I know that some kind of way that it is due to the relationship of (A-1A-1)=I

(A-1A-1) does not equal I


(A-1.A) = (A.A-1) = I

what is

(A-1.A)T ?
 
  • #3
(A.A-1) = I I am sorry I typed that wrong
 
  • #4
ok so ideas for 1) ?

for 2), what does it mean to be a matrix of rank 1? and uvT looks like an outer product, do you know how this is defined? Thinking about row reduction may help make the connection...

for 3) think about multiplying both sides of your equation by something...
 

1. What is linear transpose?

Linear transpose is a mathematical operation that involves switching the rows and columns of a matrix or vector. It is also known as matrix transposition or matrix flip.

2. Why is linear transpose important?

Linear transpose is important in various areas of science, including statistics, physics, and engineering. It allows for easier manipulation and calculation of matrices, and is used in solving systems of linear equations, finding eigenvalues and eigenvectors, and performing transformations in 3D space.

3. How is linear transpose performed?

Linear transpose is performed by taking the rows of a matrix and turning them into columns, and vice versa. This can be done by swapping the elements of the matrix across the diagonal, or by using a specific formula depending on the dimensions of the matrix.

4. What is the notation for linear transpose?

The notation for linear transpose is a superscript "T" or a prime symbol ('). For example, if A is a matrix, its transpose would be written as AT or A'.

5. What are some real-world applications of linear transpose?

Linear transpose is used in various fields such as computer graphics, data compression, signal processing, and cryptography. It is also used in machine learning algorithms, where it is used to transform data and improve the performance of models.

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