Any matrix

Any matrix $A$ has the same eigenvalues as its transpose $A^T$

Theorem-1: Any matrix $A$ has the same eigenvalues as its transpose $A^T$¹.

Proof ²³:

If $A$ is a square matrix, then its eigenvalues are equal to the eigenvalues of it transpose since they share the same characteristic polynomial.

The characteristic polynomial of $A$ is:

and for $A^T$:

Since $\vert A^T\vert=\vert A\vert$.

Q.E.D

But, it is important to highlight that, the eigenvalues of $A$ and $A^T$ are not the same, we could simply verify this point using MATLAB:

1
2
3
4
A = [1,3;2,4];

[v1,lamda1] = eig(A)
[v2,lamda2] = eig(A')

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
v1 =
   -0.9094   -0.5658
    0.4160   -0.8246

lamda1 =
   -0.3723         0
         0    5.3723

v2 =
   -0.8246   -0.4160
    0.5658   -0.9094

lamda2 =
   -0.3723         0
         0    5.3723

And actually at some special cases, the probability that $A$ and $A^T$ have no common eigenvector is $1$⁴.

Hermitian matrix

All eigenvalues of Hermitian matrix are real.

Theorem-2: All eigenvalues of Hermitian matrix are real.

Proof: The proof of this theorem could be found in⁵.

We could obtain the following corollaries from this theorem:

Corollary-2-1: All coefficients of the characteristic polynomial of Hermitian matrix are real⁵.

Corollary-2-2: The trace of Hermitian matrix is real⁵.

Corollary-2-3: The determinant of Hermitian matrix is real⁵.

Real symmetric matrix (A special Hermitian matrix)⁵

All eigenvalues of real symmetric matrix are real.

Theorem-3: All eigenvalues of real symmetric matrix are real.

Proof: According to the definition of Hermitian matrix, real symmetric matrix is a special case of Hermitian matrix⁵.

Similar to Corollary-2-1, Corollary-2-2, and Corollary-2-3 of Hermitian matrix, these corollaries are also true:

Corollary-3-1: All coefficients of the characteristic polynomial of real symmetric matrix are real.

Corollary-3-2: The trace of real symmetric matrix is real.

Corollary-3-3: The determinant of real symmetric matrix is real.

If $A$ is a real $n\times n$ symmetric matrix, the $A$ has $n$ real eigenvalues (counted by their multiplicities). And for each eigenvalue, we can find a real eigenvector associated with it.

Theorem-5: If $A$ is a real $n\times n$ symmetric matrix, the $A$ has $n$ real eigenvalues (counted by their multiplicities). And for each eigenvalue, we can find a real eigenvector associated with it.

Proof: Refer to⁶.

Eigenvectors corresponding to different eigenvalues of a real symmetric matrix are orthogonal.

Theorem-4: Eigenvectors corresponding to different eigenvalues of a real symmetric matrix are orthogonal.

Proof: The detailed proof could be found in⁶.

References

• Eigenvalues and eigenvectors.