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Let matrices $\mathbf{F}_1 \in \mathbb{R}^{n \times n}$, $\mathbf{G}_1 \in \mathbb{R}^{n\times n}$, $\mathbf{C}_1 \in \mathbb{R}^{n \times n}$, $\mathbf{D}_1 \in \mathbb{R}^{n \times n}$, $\mathbf{F}_2 \in \mathbb{R}^{n\times n}$, $\mathbf{G}_2 \in \mathbb{R}^{n\times n}$, $\mathbf{C}_2 \in \mathbb{R}^{n\times n}$ and $\mathbf{D}_2 \in \mathbb{R}^{n\times n}$ be building blocks of two larger matrices \begin{equation} \mathbf{A} = \left[\begin{array}{ccc} \mathbf{F}_1 & \mathbf{0} & \mathbf{G}_1\\ \mathbf{G}_2\mathbf{C}_1 & \mathbf{F}_2 & \mathbf{G}_2\mathbf{D}_1\\ \mathbf{D}_2\mathbf{C}_1 & \mathbf{C}_2 & \mathbf{D}_2\mathbf{D}_1 \end{array}\right] \end{equation} and \begin{equation} \mathbf{B} = \left[\begin{array}{ccc} \mathbf{F}_1 & \mathbf{G}_1\mathbf{C}_2 & \mathbf{G}_1\mathbf{D}_2\\ \mathbf{0} & \mathbf{F}_2 & \mathbf{G}_2\\ \mathbf{C}_1 & \mathbf{D}_1\mathbf{C}_2 & \mathbf{D}_1\mathbf{D}_2 \end{array}\right] \end{equation} Let $p_{\mathbf{A}(\lambda)} = \det(\lambda\mathbf{I} - \mathbf{A})$ and $p_{\mathbf{B}(\lambda)} = \det(\lambda\mathbf{I} - \mathbf{B})$ be characteristic polynomials of matrices $\mathbf{A}$ and $\mathbf{B}$, respectively. I would like to know whether the two characteristic polynomials are equal $p_{\mathbf{A}(\lambda)} = p_{\mathbf{B}(\lambda)}$.

For $n = 1$ the component matrices are scalars and $\mathbf{A},\mathbf{B} \in \mathbb{R}^{3\times 3}$. With this condition characteristic polynomials of two matrices are equal. This can be verified by the definition of the characteristic polynomial.

I don't know how to generalize this statement for $n > 1$. Any help would be appreciated.

Slaven Glumac
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1 Answers1

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Let $\mathbf{X} \in \mathbb{R}^{3n\times 3n}$ and $\mathbf{Y} \in \mathbb{R}^{3n\times 3n}$. According to Theorem 1.3.22 in the second edition of "Matrix Analysis" by Horn and Johnson characteristic polynomials of two matrix products are equal: \begin{equation} p_{\mathbf{X}\mathbf{Y}}(\lambda) = p_{\mathbf{Y}\mathbf{X}}(\lambda) \end{equation} I found the reference here.

This means if matrices $\mathbf{X}$ and $\mathbf{Y}$ exist for which $\mathbf{A} = \mathbf{X}\mathbf{Y}$ and $\mathbf{B} = \mathbf{Y}\mathbf{X}$ then characteristic polynomials of $\mathbf{A}$ and $\mathbf{B}$ are equal. These can be found by solving matrix equations: \begin{equation} \mathbf{X} = \left[\begin{array}{ccc} \mathbf{I} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{F}_2 & \mathbf{G}_2\\ \mathbf{0} & \mathbf{C}_2 & \mathbf{D}_2 \end{array}\right] \end{equation} \begin{equation} \mathbf{Y} = \left[\begin{array}{ccc} \mathbf{F}_1 & \mathbf{0} & \mathbf{G}_1\\ \mathbf{0} & \mathbf{I} & \mathbf{0}\\ \mathbf{C}_1 & \mathbf{0} & \mathbf{D}_1 \end{array}\right] \end{equation}

This proves that the statement can be generalized.

Slaven Glumac
  • 390