Some Properties of Eigenvalues and Eigenvectors

Oct. 31, 2023

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 $A^T$ [2].

Proof [3], [4]:

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:

\[\text{left}=\vert A-\lambda I\vert\]

and for $A^T$:

\[\text{right}=\vert A^T-\lambda I\vert=\vert(A-\lambda I)^T\vert=\vert A-\lambda I\vert\]

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:

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

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

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 $1$ [5].

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 in [6].

We could obtain the following corollaries from this theorem:

Corollary-2-1: All coefficients of the characteristic polynomial of Hermitian matrix are real [6].

Corollary-2-2: The trace of Hermitian matrix is real [6].

Corollary-2-3: The determinant of Hermitian matrix is real [6].

Real symmetric matrix (A special Hermitian matrix) [6]

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 matrix [6].

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 to [7: Theorem-4].