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I am preparing for an exam for which Eigen Value from linear algebra is the one of the important topic. I am listing some properties of Eigen Value that I studied:

  1. If $\lambda$ is an eigen value of a matrix $A$ then $\frac{1}{\lambda }$ will be an eigen value of inverse of Matrix $A$.

  2. If $\lambda$ is an eigen value of a matrix $A$ then $\lambda^n$ will be an eigen value of $A^n$.

  3. If $\lambda$ is an eigen value of a matrix $A$ then $k\lambda$ will be an eigen value of $kA$.

Is this list exhaustive? If not can you please suggest some more properties of Eigen Value which might be useful for my exam.

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    $\begingroup$ If $λ$ is an eigenvalue of a matrix $A$ then λ will be an eigenvalue of $A^T$ (the transpose matrix of $A$). $\endgroup$ – The Phenotype Jan 4 '18 at 15:25
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    $\begingroup$ If $\lambda$ is an eigenvalue of a matrix $A$ with $\|A\|<1$ then $1-\lambda$ will be an eigenvalue of $(I-A)$, so $\frac{1}{1-\lambda}$ will be an eigenvalue of $I+A+A^2+\ldots=(I-A)^{-1}$. $\endgroup$ – The Phenotype Jan 4 '18 at 15:31
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    $\begingroup$ Some search of yourself might me useful for your exam! I found many questions here on MSE (see below). $\endgroup$ – Dietrich Burde Jan 4 '18 at 15:36
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  1. Are the eigenvalues of $AB$ and $BA$ for two square matrices $A$,$B$ the same?

Eigenvalues of $AB$ and $BA$ the same?

  1. Are the eigenvalues of $A^TB$ and $AB^T$ the same?

Why do the eigenvalues of $A^TB$ equal the nonzero eigenvalues of $AB^T$?

  1. If $A$ and $B$ have a common eigenvalue, has $A-B$ then an eigenvalue $0$?

Eigenvalue of the substraction of 2 matrices

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  • The sum of the eigenvalues is the trace.

  • The product of the eigenvalues is the determinant.

  • An $n\times n$ matrix has $n$ eigenvalues (counted with multiplicity) $\lambda_{1},\ldots, \lambda_{n}$.

  • If $g(X)=a_{0}X^{0}+a_{1}X^{1}+\ldots +a_{m}X^{m}$ is a polynomial, then the eigenvalues of $g(A)$ (where we define $A^{0}$ as the identity) are $g(\lambda_{1}),\ldots,g(\lambda_{n})$.

  • Similar matrices (those which differ by conjugation by an invertible matrix) have the same eigenvalues.

  • The eigenvalues of a diagonal matrix are the elements in the diagonal.

Hint: you may want to prove that some equation with matrices holds to apply statements such as the fourth one and express the eigenvalues of your matrix in terms of eigenvalues of another matrix that you already know, for example. To prove an equation with matrices (with coefficients in the complex numbers), it suffices to show that the equation holds for diagonalizable matrices. This may simplify things. (The reason is that diagonalizable matrices are Zariski dense in the space of all matrices).

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The following are equivalent (in finite dimensions):

  1. $\lambda$ is an eigenvalue of $A$,
  2. $A - \lambda I$ is not injective,
  3. $A - \lambda I$ is not surjective,
  4. $A - \lambda I$ is not invertible,
  5. $\operatorname{det}(A - \lambda I) = 0$.
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