Editing Characteristic polynomial (section)

==Properties==

The characteristic polynomial <math>p_A(t)</math> of a <math>n \times n</math> matrix is monic (its leading coefficient is <math>1</math>) and its degree is <math>n.</math>  The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of <math>A</math> are precisely the [[Root of a function|root]]s of <math>p_A(t)</math> (this also holds for the [[Minimal polynomial (linear algebra)|minimal polynomial]] of <math>A,</math> but its degree may be less than <math>n</math>).  All coefficients of the characteristic polynomial are [[polynomial expression]]s in the entries of the matrix. In particular its constant coefficient of <math>t^0</math> is <math>\det(-A) = (-1)^n \det(A),</math> the coefficient of <math>t^n</math> is one, and the coefficient of <math>t^{n-1}</math> is {{math|1=tr(−''A'') = −tr(''A'')}}, where {{math|tr(''A'')}} is the [[Trace (matrix)|trace]] of <math>A.</math> (The signs given here correspond to the formal definition given in the previous section; for the alternative definition these would instead be <math>\det(A)</math> and {{math|(−1)<sup>''n'' – 1 </sup>tr(''A'')}} respectively.<ref>Theorem 4 in these [https://www.math.ucla.edu/~tao/resource/general/115a.3.02f/week8.pdf lecture notes]</ref>)

For a <math>2 \times 2</math> matrix <math>A,</math> the characteristic polynomial is thus given by
<math display=block>t^2 - \operatorname{tr}(A) t + \det(A).</math>

Using the language of [[exterior algebra]], the characteristic polynomial of an <math>n \times n</math> matrix <math>A</math> may be expressed as 
<math display=block>p_A (t) = \sum_{k=0}^n t^{n-k} (-1)^k \operatorname{tr}\left(\textstyle\bigwedge^k A\right)</math>
where <math display="inline">\operatorname{tr}\left(\bigwedge^k A\right)</math> is the [[Trace (linear algebra)|trace]] of the <math>k</math>th [[Exterior algebra#Functoriality|exterior power]] of <math>A,</math> which has dimension <math display="inline">\binom {n}{k}.</math> This trace may be computed as the sum of all [[principal minor]]s of <math>A</math> of size <math>k.</math> The recursive [[Faddeev–LeVerrier algorithm]] computes these  coefficients  more efficiently {{Clarify|reason=More efficiently than what or who?|date=March 2025}}.

When the [[Characteristic (algebra)|characteristic]] of the [[Field (mathematics)|field]] of the coefficients is <math>0,</math> each such trace may alternatively be computed as a single determinant, that of the <math>k \times k</math> matrix,
<math display=block>\operatorname{tr}\left(\textstyle\bigwedge^k A\right) = \frac{1}{k!}  
\begin{vmatrix}  \operatorname{tr}A  &   k-1 &0&\cdots &0 \\
\operatorname{tr}A^2  &\operatorname{tr}A&  k-2 &\cdots &0 \\
 \vdots & \vdots & & \ddots & \vdots    \\
\operatorname{tr}A^{k-1} &\operatorname{tr}A^{k-2}& & \cdots & 1    \\ 
\operatorname{tr}A^k  &\operatorname{tr}A^{k-1}& & \cdots & \operatorname{tr}A
\end{vmatrix} ~.</math>

The [[Cayley–Hamilton theorem]] states that replacing <math>t</math> by <math>A</math> in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term <math>c</math> as <math>c</math> times the identity matrix) yields the zero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the [[Minimal polynomial (linear algebra)|minimal polynomial]] of <math>A</math> divides the characteristic polynomial of <math>A.</math>

Two [[similar matrices]] have the same characteristic polynomial.  The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.

The matrix <math>A</math> and its [[transpose]] have the same characteristic polynomial. <math>A</math> is similar to a [[triangular matrix]] [[if and only if]] its characteristic polynomial can be completely factored into linear factors over <math>K</math> (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case <math>A</math> is similar to a matrix in [[Jordan normal form]].