# Q1: Does A and its transpose have the same eigenspace?

• Howers
In summary, A and A^T have the same characteristic polynomials and therefore the same eigenvalues due to the fact that detA = detA^T. However, the eigenspaces, or sets of eigenvectors and eigenrows, will not necessarily be the same unless A is a symmetric matrix. This is because there are two types of eigenvectors, right and left, and they are only equal when the set of right eigenvectors is orthonormal. It is possible to select specially scaled sets of eigenvectors and eigenrows to make them coincide, but this is not always the case.
Howers
So I've shown that A and A^T have the same char. polynomials => same eigenvalues, using the fact that detA = detA^T. I still can't see any way I could possibly show or disprove that the eigenspace is the same.

$$\left(\begin{array}{ll} 1 & 1\\ 0 & 0 \end{array}\right)\ \ \left(\begin{array}{ll} 1 & 0\\ 1 & 0 \end{array}\right)$$
Do they have the same eigenspaces?

A and A^T will not have the same eigenspaces, i.e. eigenvectors, in general.

Remember that there are in fact two "eigenvectors" for every eigenvalue $$\lambda$$. The right eigenvector satisfying $$A\mathbf{x} = \lambda \mathbf{x}$$ and a left eigenvector (eigenrow?) satisfying $$\mathbf{x}A = \lambda \mathbf{x}$$. In general these are not equal.

Also, I believe that the set of left eigenvectors is the inverse matrix of the set of right eigenvectors, but I am not about sure of this. If this is indeed the case then the set of left eigenvectors will "coincide" with the set of right eigenvectors only when the set of right eigenvectors is orthonormal, i.e. when A is symmetric A=A^T.

EDIT: In fact, the conjecture above is not true unless you select specially scaled sets of eigenvectors and eigenrows. Is there a way of selecting eigenvectors of canonical lengths?

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## 1. What is an eigenspace?

An eigenspace is a vector space associated with a particular eigenvalue of a matrix. It consists of all the eigenvectors corresponding to that eigenvalue, as well as the zero vector.

## 2. What is the transpose of a matrix?

The transpose of a matrix is a new matrix that is obtained by flipping the rows and columns of the original matrix. This is denoted by adding a superscript "T" to the original matrix.

## 3. How do you determine if two matrices have the same eigenspace?

To determine if two matrices have the same eigenspace, you need to compare their eigenvalues and eigenvectors. If the eigenvalues are the same and the corresponding eigenvectors are linearly dependent, then the matrices have the same eigenspace.

## 4. What is the significance of having the same eigenspace for A and its transpose?

If A and its transpose have the same eigenspace, it means that they have the same eigenvalues and eigenvectors. This can be useful in certain applications, such as in solving systems of linear equations or in diagonalizing a matrix.

## 5. Can two matrices have the same eigenspace even if they are not equal?

Yes, it is possible for two matrices to have the same eigenspace even if they are not equal. This is because the eigenspace is determined by the eigenvalues and eigenvectors, which can be the same for different matrices even if the matrices themselves are not equal.

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