Easy Way to Find the Characteristic Polynomial of a 5x5 Matrix
Tool to calculate the characteristic polynomial of a matrix. The characteristic polynomial of a matrix M is computed as the determinant of (X.I-M).
Characteristic Polynomial of a Matrix - dCode
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Characteristic Polynomial of a Matrix
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- Matrix
- Characteristic Polynomial of a Matrix
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Answers to Questions (FAQ)
What is the characteristic polynomial for a matrix? (Definition)
The characteristic polynomial (or sometimes secular function) $ P $ of a square matrix $ M $ of size $ n \times n $ is the polynomial defined by $$ P_M(x) = \det(M - x.I_n) \tag{1} $$ or $$ P_M(x) = \det(x.I_n - M) \tag{2} $$ with $ I_n $ the identity matrix of size $ n $ (and det the matrix determinant).
The 2 possible values $ (1) $ and $ (2) $ give opposite results, but since the polynomial is used to find roots, the sign does not matter.
The equation $ P = 0 $ is called the characteristic equation of the matrix.
Why calculating the characteristic polynomial of a matrix?
The characteristic polynomial $ P $ of a matrix, as its name indicates, characterizes a matrix, it allows in particular to calculate the eigenvalues and the eigenvectors.
How to calculate the characteristic polynomial of a diagonal matrix?
If $ M $ is a diagonal matrix with $ \lambda_1, \lambda_2, \ldots, \lambda_n $ as diagonal elements, then the computation is simplified and $$ P_M(x) = (x-\lambda_1)(x-\lambda_2)\ldots(x-\lambda_n) $$
How to calculate the characteristic polynomial of a triangualr matrix?
If $ M $ is a triangular matrix with $ \lambda_1, \lambda_2, \ldots, \lambda_n $ as diagonal elements, then as for diagonal matrix, the computation is simplified and $$ P_M(x) = (x-\lambda_1)(x-\lambda_2)\ldots(x-\lambda_n) $$
How to calculate the characteristic polynomial for a 2x2 matrix?
The calculation of the characteristic polynomial of a square matrix of order 2 can be calculated with the determinant of the matrix $ [ x.I_2 - M ] $ as $$ P_M(x) = \det [ x.I_2 - M ] $$
The polynomial can also be written with another formula using the trace of the matrix $ M $ (noted Tr): $$ P_{M_2}(x) = \det( x.I_2 - M ) = x^2 - \operatorname{Tr}(M)x+ \det(M) $$
Example: $$ M=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \\ \Rightarrow x.I_n - M = \begin{bmatrix} x & 0 \\ 0 & x \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} x-1 & -2 \\ -3 & x-4 \end{bmatrix} \\ \Rightarrow \det(x.I_n - M) = (x-1)(x-4)-((-2)\times(-3)) \\ \Rightarrow P_M(x) = x^2-5x-2 $$
How to calculate the characteristic polynomial for a 3x3 matrix?
Calculation of the characteristic polynomial of a square 3x3 matrix can be calculated with the determinant of the matrix $ [ x.I_3 - M ] $ as $$ P_M(x) = \det [ x.I_3 - M ] $$
Example: $$ M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} $$ $$ [ x.I_3 - M ] = x \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} - M = \begin{bmatrix} x-a & -b & -c \\ -d & x-e & -f \\ -g & -h & x-i \end{bmatrix} $$ $$ P_M(x) = \det [ x.I_3 - M ] = -a e i+a e x+a f h+a i x-a x^2+b d i-b d x-b f g-c d h+c e g-c g x+e i x-e x^2-f h x-i x^2+x^3 $$
It is also possible to use another formula with the Trace of the matrix $ M $ (noted Tr): $$ P_{M_3}(x) = -x^3 + \operatorname{Tr}(M)x^2 + \frac{1}{2} \left( \operatorname{Tr}^2(M) - \operatorname{Tr}(M^2) \right) x + \frac{1}{6} \left( \operatorname{Tr}^3(M) + 2\operatorname{Tr}(M^3) - 3\operatorname{Tr}(M)\operatorname{Tr}(M^2) \right) $$
How to calculate the characteristic polynomial for a 4x4 matrix?
Calculation of the characteristic polynomial of an order 4 square matrix can be calculated with the determinant of the matrix $ [ x.I_4 - M ] $ as $$ P_M(x) = \det [ x.I_4 - M ] $$
Example: $$ M = \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{bmatrix} $$ $$ [ x.I_4 - M ] = x \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} - M = \begin{bmatrix} x-a & b & c & d \\ e & x-f & g & h \\ i & j & x-k & l \\ m & n & o & x-p \end{bmatrix} $$ $$ P_M(x) = \det [ x.I_4 - M ] = a f k p-a f k x-a f l o-a f p x+a f x^2-a g j p+a g j x+a g l n+a h j o-a h k n+a h n x-a k p x+a k x^2+a l o x+a p x^2-a x^3-b e k p+b e k x+b e l o+b e p x-b e x^2+b g i p-b g i x-b g l m-b h i o+b h k m-b h m x+c e j p-c e j x-c e l n-c f i p+c f i x+c f l m+c h i n-c h j m+c i p x-c i x^2-c l m x-d e j o+d e k n-d e n x+d f i o-d f k m+d f m x-d g i n+d g j m-d i o x+d k m x-d m x^2-f k p x+f k x^2+f l o x+f p x^2-f x^3+g j p x-g j x^2-g l n x-h j o x+h k n x-h n x^2+k p x^2-k x^3-l o x^2-p x^3+x^4 $$
Is there multiple characteristic polynomials for a matrix?
The characteristic polynomial is unique for a given matrix. There is only one way to calculate it and it has only one result.
On the other hand, two different matrices can give the same characteristic polynomial.
How to calculate the characteristic polynomial for a transpose matrix?
A matrix $ M $ and its matrix transpose $ M^T $ have the same characteristic polynomial.
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