How to Prove Rank(A) Equals Rank(ATA)?

In summary, proving that the rank of A is equal to the rank of ATA is significant in understanding the properties and structure of a matrix. It provides insight into linear independence, dimension, and nullspace, which are crucial in applications such as data analysis and optimization problems. The rank of A and the rank of ATA are equal because they both represent the number of linearly independent columns in a matrix. To prove this, we can use the fact that the rank of a matrix is equal to the number of non-zero singular values. However, it is not always true that the rank of A is equal to the rank of ATA. The rank of A is equal to the dimension of the column space of A, which represents the number of linearly
  • #1
katyyara
3
0
Hi,

Does anyone know how to prove rank(A)=rank(AT A) where A is any matrix and AT is the transposed of matrix A? I have difficulty to prove the part that nulity(A)=nulity(AT A). Any help will be appreciated.
 
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  • #2
Try proving [tex]\operatorname{ker} A = \operatorname{ker} ( A^\top A)[/tex].
 

FAQ: How to Prove Rank(A) Equals Rank(ATA)?

What is the significance of proving that the rank of A is equal to the rank of ATA?

Proving that the rank of A is equal to the rank of ATA is important in understanding the properties and structure of a matrix. It provides insight into the linear independence, dimension, and nullspace of a matrix, which are crucial in applications such as data analysis and optimization problems.

Can you explain the relationship between the rank of A and the rank of ATA?

The rank of A and the rank of ATA are equal because they both represent the number of linearly independent columns in a matrix. This is due to the fact that ATA is the product of A and its transpose, which preserves the linear independence of columns.

How do you prove that the rank of A is equal to the rank of ATA?

To prove that the rank of A is equal to the rank of ATA, we can use the fact that the rank of a matrix is equal to the number of non-zero singular values. Therefore, by showing that A and ATA have the same singular values, we can conclude that their ranks are also equal.

Is it always true that the rank of A is equal to the rank of ATA?

No, it is not always true. The rank of A is equal to the rank of ATA if and only if A is a square matrix or if A is a rectangular matrix with full column rank, meaning all its columns are linearly independent. In other cases, the rank of A may be less than the rank of ATA.

How is the rank of A related to the dimension of the column space of A?

The rank of A is equal to the dimension of the column space of A. This means that the rank of A represents the number of linearly independent columns in A, which is the same as the dimension of the column space. The column space is the set of all possible linear combinations of the columns of A, and its dimension is equal to the number of linearly independent columns.

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