In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial or secular equation. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace (matrix). ==Motivation== Given a square matrix ''A'', we want to find a polynomial whose roots are precisely the eigenvalues of ''A''. For a diagonal matrix ''A'', the characteristic polynomial is easy to define: if the diagonal entries are ''a'', ''b'', ''c'' the characteristic polynomial will be :(''t'' − ''a'' )(''t'' − ''b'' )(''t'' − ''c'' )... up to a convention about sign (+ or -). This works because the diagonal entries are also the eigenvalues of this matrix. For a general matrix ''A'', one can proceed as follows. If λ is an eigenvalue of ''A'', then there is an eigenvector v≠0 such that :''A'' v = λv, or :(λ''I'' - ''A'' )v = 0 (where ''I'' is the identity matrix). Since v is non-zero, this means that the matrix λ''I'' - ''A'' is singular matrix, which in turn means that its determinant is 0. We have just shown that the roots of the function determinant(''t'' ''I'' - ''A'') are the eigenvalues of ''A''. Since this function is a polynomial in ''t'', we're done. ==Formal definition== We start with a field (mathematics) ''K'' (you can think of ''K'' as the real number or complex number numbers) and an ''n''×''n'' matrix ''A'' over ''K''. The characteristic polynomial of ''A'', denoted by ''p''''A''(''t'' ), is the polynomial defined by :''p''''A''(''t'' ) = det(''t'' ''I'' - ''A'' ) where ''I'' denotes the ''n''-by-''n'' identity matrix. This is indeed a polynomial, since determinants are defined in terms of sums of products. (Some authors define the characteristic polynomial to be det(''A'' - ''t'' ''I'' ); the difference is immaterial since the two polynomials differ at most by a sign.) ==Example== Suppose we want to compute the characteristic polynomial of the matrix : We have to compute the determinant of : and this determinant is : The latter is the characteristic polynomial of ''A''. ==Properties== The polynomial ''p''''A''(''t'' ) is monic (its leading coefficient is 1) and its degree is ''n''. The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of ''A'' are precisely the roots of ''p''''A''(''t'' ). The constant coefficient ''p''''A''(0) is equal to (-1)''n'' times the determinant of ''A'', and the coefficient of ''t'' ''n'' -1 is equal to the negative of the trace (matrix) of ''A''. For a 2×2 matrix ''A'', the characteristic polynomial is nicely expressed then as : ''t'' 2 - tr(''A'')''t'' + det(''A'') where tr(''A'') represents the matrix trace of ''A'' and det(''A'') the determinant of ''A''. The Cayley-Hamilton theorem states that replacing ''t'' by ''A'' in the expression for ''p''''A''(''t'' ) yields the zero matrix: ''p''''A''(''A'' ) = 0. Simply, every matrix satisfies its own characteristic equation. As a consequence of this, one can show that the minimal polynomial of ''A'' divides the characteristic polynomial of ''A''. Two similarity (mathematics) 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 ''A'' and its transpose have the same characteristic polynomial. ''A'' is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over ''K''. In fact, ''A'' is even similar to a matrix in Jordan normal form in this case. Polynomials Linear algebra
This description gets a bit murky to the non-mathematician at the definition of the polynomial. What is t? What is a "polynomial ring"? Is there a simpler way to describe this concept without bringing in so many other mathematical areas, or at least a way to make them optional? Perhaps a good example is required to help ground the definition, although that might damage the generality. User:Brent Gulanowski 19:01, 30 Nov 2003 (UTC) Well, it's dense rather than murky. But I agree, really. I've added some initial comments that are intended to clarify what is happening. We'll see if these are to others' taste. User:Charles Matthews 17:22, 1 Dec 2003 (UTC) Well, I'm sorry to say that I wasn't too happy with the motivational part, since it didn't really say what the goal was (to get a polynomial whose zeros are the eigenvalues) and it used the fact that every matrix can be approximated by diagonalizable ones, which is not intuitively clear. I tried to write some other motivational intro. Also I added an example and I changed the definition to det(tI-A), since that is a monic polynomial and it works better with the companion matrix article. User:AxelBoldt 18:45, 11 Jul 2004 (UTC) The approximation business - it may ''not'' be intuitive, but it certainly helps a great deal to understand linear algebra if one has this concept. I was once told that my proof of the Cayley-Hamilton theorem using it was the 'worst ever'. But that was by a functional-analyst; while to an algebraic geometer it is just a good way to use the Zariski topology, and then use the fact that identities hold on ''closed'' sets. So, I wonder where it belongs in the WP articles. User:Charles Matthews 07:34, 15 Jul 2004 (UTC) ==Roots and zeroes== I think this is a somewhat pedantic point. But given the edit comment ''The values for which a polynomial has a value of zero are called 'roots' and not 'zeros'', I think it should be pointed out that in P. M. Cohn's ''Algebra'', it is polynomials that have zeroes and equations P = 0 that have roots. User:Charles Matthews 09:10, 15 Jul 2004 (UTC) == another 'WP' style formal definition == Once again, a page starting with "In [ some area of mathematics ]..." with a "Formal definition" going like "...you can think of..." and "This [ det(tI-A) ] is indeed a polynomial, since determinants are defined in terms of sums of products." * Nowhere is said what ''t'' is and how it can multiply ''I'' * I don't even dare to ask what the author calls a polynomial (imho, "in mathematics" (*sigh*), this should be a map from N into some group) * same question for determinants (at least, I know another definition than the above...) Now, if you accept this definition, then you also must accept the usual "student's proof" of the Cayley-Hamilton theorem: : P(A) = det( AI-A ) = det( A-A ) = det O = 0. Easy ! What's all that fuzz about ? User:MFH— User:MFH: User talk:MFH 22:41, 24 May 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Characteristic_polynomial:
Characteristic_polynomial
Characteristic_polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial or secular equation. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace (matrix). ==Motivation== Given a square matrix ''A'', we want to find a polynomial whose roots are precisely the eigenvalues of ''A''. For a diagonal matrix ''A'', the characteristic polynomial is easy to define: if the diagonal entries are ''a'', ''b'', ''c'' the characteristic polynomial will be :(''t'' − ''a'' )(''t'' − ''b'' )(''t'' − ''c'' )... up to a convention about sign (+ or -). This works because the diagonal entries are also the eigenvalues of this matrix. For a general matrix ''A'', one can proceed as follows. If λ is an eigenvalue of ''A'', then there is an eigenvector v≠0 such that :''A'' v = λv, or :(λ''I'' - ''A'' )v = 0 (where ''I'' is the identity matrix). Since v is non-zero, this means that the matrix λ''I'' - ''A'' is singular matrix, which in turn means that its determinant is 0. We have just shown that the roots of the function determinant(''t'' ''I'' - ''A'') are the eigenvalues of ''A''. Since this function is a polynomial in ''t'', we're done. ==Formal definition== We start with a field (mathematics) ''K'' (you can think of ''K'' as the real number or complex number numbers) and an ''n''×''n'' matrix ''A'' over ''K''. The characteristic polynomial of ''A'', denoted by ''p''''A''(''t'' ), is the polynomial defined by :''p''''A''(''t'' ) = det(''t'' ''I'' - ''A'' ) where ''I'' denotes the ''n''-by-''n'' identity matrix. This is indeed a polynomial, since determinants are defined in terms of sums of products. (Some authors define the characteristic polynomial to be det(''A'' - ''t'' ''I'' ); the difference is immaterial since the two polynomials differ at most by a sign.) ==Example== Suppose we want to compute the characteristic polynomial of the matrix : We have to compute the determinant of : and this determinant is : The latter is the characteristic polynomial of ''A''. ==Properties== The polynomial ''p''''A''(''t'' ) is monic (its leading coefficient is 1) and its degree is ''n''. The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of ''A'' are precisely the roots of ''p''''A''(''t'' ). The constant coefficient ''p''''A''(0) is equal to (-1)''n'' times the determinant of ''A'', and the coefficient of ''t'' ''n'' -1 is equal to the negative of the trace (matrix) of ''A''. For a 2×2 matrix ''A'', the characteristic polynomial is nicely expressed then as : ''t'' 2 - tr(''A'')''t'' + det(''A'') where tr(''A'') represents the matrix trace of ''A'' and det(''A'') the determinant of ''A''. The Cayley-Hamilton theorem states that replacing ''t'' by ''A'' in the expression for ''p''''A''(''t'' ) yields the zero matrix: ''p''''A''(''A'' ) = 0. Simply, every matrix satisfies its own characteristic equation. As a consequence of this, one can show that the minimal polynomial of ''A'' divides the characteristic polynomial of ''A''. Two similarity (mathematics) 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 ''A'' and its transpose have the same characteristic polynomial. ''A'' is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over ''K''. In fact, ''A'' is even similar to a matrix in Jordan normal form in this case. Polynomials Linear algebra
Characteristic polynomial
This description gets a bit murky to the non-mathematician at the definition of the polynomial. What is t? What is a "polynomial ring"? Is there a simpler way to describe this concept without bringing in so many other mathematical areas, or at least a way to make them optional? Perhaps a good example is required to help ground the definition, although that might damage the generality. User:Brent Gulanowski 19:01, 30 Nov 2003 (UTC) Well, it's dense rather than murky. But I agree, really. I've added some initial comments that are intended to clarify what is happening. We'll see if these are to others' taste. User:Charles Matthews 17:22, 1 Dec 2003 (UTC) Well, I'm sorry to say that I wasn't too happy with the motivational part, since it didn't really say what the goal was (to get a polynomial whose zeros are the eigenvalues) and it used the fact that every matrix can be approximated by diagonalizable ones, which is not intuitively clear. I tried to write some other motivational intro. Also I added an example and I changed the definition to det(tI-A), since that is a monic polynomial and it works better with the companion matrix article. User:AxelBoldt 18:45, 11 Jul 2004 (UTC) The approximation business - it may ''not'' be intuitive, but it certainly helps a great deal to understand linear algebra if one has this concept. I was once told that my proof of the Cayley-Hamilton theorem using it was the 'worst ever'. But that was by a functional-analyst; while to an algebraic geometer it is just a good way to use the Zariski topology, and then use the fact that identities hold on ''closed'' sets. So, I wonder where it belongs in the WP articles. User:Charles Matthews 07:34, 15 Jul 2004 (UTC) ==Roots and zeroes== I think this is a somewhat pedantic point. But given the edit comment ''The values for which a polynomial has a value of zero are called 'roots' and not 'zeros'', I think it should be pointed out that in P. M. Cohn's ''Algebra'', it is polynomials that have zeroes and equations P = 0 that have roots. User:Charles Matthews 09:10, 15 Jul 2004 (UTC) == another 'WP' style formal definition == Once again, a page starting with "In [ some area of mathematics ]..." with a "Formal definition" going like "...you can think of..." and "This [ det(tI-A) ] is indeed a polynomial, since determinants are defined in terms of sums of products." * Nowhere is said what ''t'' is and how it can multiply ''I'' * I don't even dare to ask what the author calls a polynomial (imho, "in mathematics" (*sigh*), this should be a map from N into some group) * same question for determinants (at least, I know another definition than the above...) Now, if you accept this definition, then you also must accept the usual "student's proof" of the Cayley-Hamilton theorem: : P(A) = det( AI-A ) = det( A-A ) = det O = 0. Easy ! What's all that fuzz about ? User:MFH— User:MFH: User talk:MFH 22:41, 24 May 2005 (UTC)
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