Linear algebra identities of inverse matricies/transpose

In summary, the conversation discusses left and right inversion in rectangular cases for a matrix A. The left-inverse of A is defined as (A^T * A)^(-1) * A^T and can be used to show that A^(-1)_left * A = I if A has a pivot in every column. Similarly, the right-inverse of A is defined as A^T * (A * A^T)^(-1) and can be used to show that A * A^(-1)_right = I if A has a pivot in every row. The concept of left and right inverse is important in linear algebra and can be used to find solutions to systems of equations.
  • #1
SpiffyEh
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0

Homework Statement


Left Inversion in Rectangular Cases. Let A[tex]^{-1}_{left}[/tex] = (A[tex]^{T}[/tex]A)[tex]^{-1}[/tex]A[tex]^{T}[/tex] show A[tex]^{-1}_{left}[/tex]A = I.

This matrix is called the left-inverse of A and it can be shown that if A [tex]\in[/tex] R[tex]^{m x n}[/tex] such that A has a pivot in every column then the left inverse exists.

Right Inversion in Rectangular Cases. Let A[tex]^{-1}_{right}[/tex] = A[tex]^{T}[/tex](AA[tex]^{T}[/tex])[tex]^{-1}[/tex]. Show AA[tex]^{-1}_{right}[/tex] = I.

This matrix is called the right-inverse of A and it can be shown that if A [tex]\in[/tex] R[tex]^{m x n}[/tex] such that A has a pivot in every row then the right inverse exists.

Homework Equations





The Attempt at a Solution


I tried the left part and this is what I did:
A[tex]^{-1}_{left}[/tex] / (A[tex]^{T}[/tex]A)[tex]^{-1}[/tex]= A[tex]^{T}[/tex]
A[tex]^{-1}_{left}[/tex](A[tex]^{T}[/tex]A) = A[tex]^{T}[/tex]
A[tex]^{-1}_{left}[/tex]A = A[tex]^{T}[/tex]( A[tex]^{T}[/tex])[tex]^{-1}[/tex] = I

I'm not sure if this is correct or not, so I want to see if I have the right idea. I know that A*A[tex]^{-1}[/tex] = I so I thought this would work. Also isn't the right one the exact same thing? Or do I have to do that one a different way? Oh and can someone also explain the concept of left and right inverse. I don't really understand it. Thanks
 
Last edited:
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  • #2
Sorry it's not showing right at all. I'll try to make it clearer. Al is A[tex]_{left}[/tex].

Attempt:
Al^(-1) / (A^T * A)^(-1) = A^T
Al^(-1) * (A^T * A) = A^T
Al^(-1) * A = A^T * (A^T)^(-1) = I
therefore, Al^(-1) * A = I

Hopefully that made what I was trying to do more clear

Can someone please help?
 
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Related to Linear algebra identities of inverse matricies/transpose

1. What is a linear algebra identity?

A linear algebra identity is a mathematical equation that holds true for all values of the variables involved. It is a fundamental concept in linear algebra and is used to prove theorems and solve problems.

2. What is an inverse matrix?

An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is denoted as A-1 and is used to solve linear systems of equations and perform operations such as matrix division.

3. How do you find the inverse of a matrix?

The inverse of a matrix can be found by using the Gauss-Jordan elimination method or by using the adjugate matrix method. Both methods involve performing a series of elementary row operations on the original matrix until it is transformed into the identity matrix.

4. What is a transpose matrix?

A transpose matrix is a matrix in which the rows and columns are switched. So, if the original matrix has dimensions m x n, the transpose matrix will have dimensions n x m. It is denoted as AT and is used to perform operations such as matrix multiplication and solving systems of equations.

5. How are linear algebra identities of inverse matrices and transpose matrices related?

The transpose of the inverse of a matrix is equal to the inverse of the transpose of that matrix. In other words, (A-1)T = (AT)-1. This identity is useful in solving equations involving inverse and transpose matrices, as well as in proving other linear algebra identities.

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