Prove AtA is Nonsingular: Invertible Matrix Homework

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In summary, if the columns of a matrix A are linearly independent, then the matrix A^(T)A is nonsingular. This can be proven by using the fact that it is sufficient to show that null(A^(T)A) = {0}, and since the columns of A are linearly independent, null(A) = {0}. This means that A^(T)A cannot be singular, as it would result in a contradiction.
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roam
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Homework Statement



If A is a an mxn matrix and its column vectors are linearly independent.

Prove that the matrix AtA is nonsingular. Hint: Use the fact that it is sufficient to show that null(AtA) = {0}

Homework Equations





The Attempt at a Solution



I'm new to this topic & I don't understand the hint given and how exactly to use it to prove the question...

I know that in order for a matrix to be nonsingular/invertible it has to be squre (m=n) and when you multiply the a matrix by its transpose, the resulting matrix would be square.
I'm also thinking about the properties of fundamental spaces of matrices that: row(A) = null(A) and col(A)=null(At) (therefore null(AtA) = row(A).col(A)?)

Any help would be much appreciated :)

Cheers.
 
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  • #2
If the columns of A are linearly independent, doesn't that mean null(A)={0}? Now suppose A^(T)A were singular and think about x^(T)A^(T)*Ax, where x is a column vector in R^n.
 

1. What does it mean for a matrix to be nonsingular?

For a matrix to be nonsingular means that it is invertible, or it has an inverse matrix. This means that the matrix can be multiplied by another matrix to yield the identity matrix, or the matrix with 1's along the diagonal and 0's everywhere else.

2. How can you prove that a matrix is nonsingular?

One way to prove that a matrix is nonsingular is by showing that its determinant is non-zero. If the determinant is equal to 0, then the matrix is singular and does not have an inverse.

3. What is the inverse of a matrix?

The inverse of a matrix is another matrix that, when multiplied together, yields the identity matrix. In other words, it "undoes" the original matrix.

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

To find the inverse of a matrix, you can use the Gauss-Jordan elimination method or use the formula for finding the inverse of a 2x2 matrix. For larger matrices, it is often easier to use a calculator or computer program to find the inverse.

5. Why is proving that a matrix is nonsingular important?

Proving that a matrix is nonsingular is important because it ensures that the matrix is invertible. This is necessary for certain mathematical operations and can also be used to solve systems of linear equations. Additionally, nonsingular matrices have special properties and are often used in applications like computer graphics and engineering.

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