For a square matrix use the monic convention $p_A(t)=\det(tI-A)$. Its degree is the matrix size, and its roots are eigenvalues. The minimal polynomial $m_A$ is the unique monic polynomial of least degree annihilating $A$. It divides $p_A$ and has the same roots over a splitting field, with possibly smaller multiplicities.
For $A=cI_n$, $p_A(t)=(t-c)^n$ but $m_A(t)=t-c$. Splitting into distinct linear factors characterizes diagonalizability for the minimal polynomial.
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- Definition of applying a polynomial to a square matrixFor $p(t)=a 0+a 1t+\cdots+a kt^k$ and square $A$, define $p(A)=a 0I+a 1A+\cdots+a kA^k$. The constant term uses $I$,...
- Definition of characteristic polynomial of a matrixUse the monic convention $p A(t)=\det(tI-A)$ for an $n\times n$ matrix. It is a degree-$n$ polynomial whose roots are...
- Definition of characteristic equation of a matrixThe characteristic equation is $\det(tI-A)=0$. Its solutions in the chosen field are eigenvalues. For...
- Definition of minimal polynomial of a matrixThe minimal polynomial $m A$ is the unique monic polynomial of smallest degree satisfying $m A(A)=0$. It divides...
- The minimal polynomial of a square matrix exists and is unique.The powers $I,A,A^2,\ldots$ lie in the finite-dimensional space of matrices and are eventually linearly dependent, so...
- The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.A scalar $\lambda$ is an eigenvalue iff $A-\lambda I$ has nonzero kernel, iff it is singular, iff $\det(\lambda...
- The characteristic polynomial applied to the matrix gives the 0 matrix.The Cayley–Hamilton identity is $p A(A)=0$, where $p A(t)=\det(tI-A)$. Example: for...
- The Cayley-Hamilton theorem for a matrix.Cayley–Hamilton theorem: every square matrix satisfies its characteristic polynomial, $p A(A)=0$. One proof expands...
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