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Math Help - Trace = Sum of eigen values - proof?

  1. #1
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    Trace = Sum of eigen values - proof?

    Hi,

    I recently learnt that trace (sum of diagonal elements) of a matrix is equal to the sum of its eigen values. I was wondering why this holds true. Could anyone explain (I do not necessarily need a rigorous proof - concept by means of geometry, or what is happening in a vector space where all the columns are vectors etc would do too) ?

    Thank you!
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    Quote Originally Posted by Theju View Post
    Hi,

    I recently learnt that trace (sum of diagonal elements) of a matrix is equal to the sum of its eigen values. I was wondering why this holds true. Could anyone explain (I do not necessarily need a rigorous proof - concept by means of geometry, or what is happening in a vector space where all the columns are vectors etc would do too) ?

    Thank you!
    If A is a diagnolizable matrix then it is similar to a diagnol matrix consisting of its eigenvalues in the diagnols. But the trace of similar matrices is similar. Therefore the trace is the sum of its eigenvalues.
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  3. #3
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    Quote Originally Posted by Theju View Post
    Hi,

    I recently learnt that trace (sum of diagonal elements) of a matrix is equal to the sum of its eigen values. I was wondering why this holds true. Could anyone explain (I do not necessarily need a rigorous proof - concept by means of geometry, or what is happening in a vector space where all the columns are vectors etc would do too) ?

    Thank you!
    we know that the sum of zeros of a polynomial f(x)=x^n + c_1x^{n-1} + \cdots + c_n is -c_1. now the eigenvalues of a matrix A are the zeros of the polynomial p(\lambda)=\det(\lambda I-A). so we only need

    to prove that the coefficient of \lambda^{n-1} in p(\lambda) is equal to -\text{tr}(A). this can be easily proved: if A=[a_{ij}] is an n \times n matrix, then:

    p(\lambda) = \det(\lambda I - A)= \begin{vmatrix} \lambda - a_{11} & -a_{12} & \cdots & -a_{1n} \\ -a_{21} & \lambda - a_{22} & \cdots & -a_{2n} \\ . & . & \cdots & . \\ . & . & \cdots & . \\ . & . & \cdots & . \\ -a_{n1} & -a_{n2} & \cdots & \lambda - a_{nn} \end{vmatrix}. it's easy to see that in the expansion of this determinant, all terms are polynomials (in \lambda) of degree at most n-2 except for the term

    q(\lambda)=(\lambda -a_{11})(\lambda - a_{22}) \cdots (\lambda - a_{nn}). thus the coefficient of \lambda^{n-1} in p(\lambda) is equal to the coefficient of \lambda^{n-1} in q(\lambda), which clearly is: -(a_{11}+a_{22} + \cdots + a_{nn})= - \text{tr}(A). \ \ \ \Box
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    Hi,

    Thank you for your responses. They answered my questions.
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  5. #5
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    thank you for your help it has saved alot of my time, eanbling me to solve this quest

    Quote Originally Posted by NonCommAlg View Post
    we know that the sum of zeros of a polynomial f(x)=x^n + c_1x^{n-1} + \cdots + c_n is -c_1. now the eigenvalues of a matrix A are the zeros of the polynomial p(\lambda)=\det(\lambda I-A). so we only need

    to prove that the coefficient of \lambda^{n-1} in p(\lambda) is equal to -\text{tr}(A). this can be easily proved: if A=[a_{ij}] is an n \times n matrix, then:

    p(\lambda) = \det(\lambda I - A)= \begin{vmatrix} \lambda - a_{11} & -a_{12} & \cdots & -a_{1n} \\ -a_{21} & \lambda - a_{22} & \cdots & -a_{2n} \\ . & . & \cdots & . \\ . & . & \cdots & . \\ . & . & \cdots & . \\ -a_{n1} & -a_{n2} & \cdots & \lambda - a_{nn} \end{vmatrix}. it's easy to see that in the expansion of this determinant, all terms are polynomials (in \lambda) of degree at most n-2 except for the term

    q(\lambda)=(\lambda -a_{11})(\lambda - a_{22}) \cdots (\lambda - a_{nn}). thus the coefficient of \lambda^{n-1} in p(\lambda) is equal to the coefficient of \lambda^{n-1} in q(\lambda), which clearly is: -(a_{11}+a_{22} + \cdots + a_{nn})= - \text{tr}(A). \ \ \ \Box
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    linear algebra

    i really don't know how to solve this quest

    a^2+b^2+c^2=d^2+e^2=365
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  7. #7
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    What, exactly, is the question? You have essentially two equations, one in three unknowns, the other for two unknowns. They can't be solved for specific numbers.

    By the way, you have committed three major sins here:
    1) you posted a question that made no sense.

    2) you posted a question in "Linear Algebra and Abstract Algebra" that has nothing to do with either.

    3) you "hijacked" someone else's thread to ask a completely unrelated question.
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  8. #8
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    Re: Trace = Sum of eigen values - proof?

    Let A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}

    Define E = \begin{bmatrix} \lambda I - A\end{bmatrix} = \begin{bmatrix} \lambda - a_{11} & -a_{12} & \cdots & -a_{1n} \\ -a_{21} & \lambda - a_{22} & \cdots & -a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{n1} & -a_{n2} & \cdots & \lambda - a_{nn} \end{bmatrix} = \begin{bmatrix} e_{11} & e_{12} & \cdots & e_{1n} \\ e_{21} & e_{22} & \cdots & e_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ e_{n1} & e_{n2} & \cdots & e_{nn} \end{bmatrix}

    Note e_{ij} contains \lambda, iff, e_{ij} is an entry in Es diagonal (e_{ij} satisfies i = j).

    Set \chi(\lambda) = \begin{vmatrix} E \end{vmatrix} = \begin{vmatrix} \lambda I - A \end{vmatrix} = \lambda^n + c_1\lambda^{n-1} + \cdots + c_n. Applying the Fundamental Theorem of Algebra to \chi(\lambda) repeatedly, we have \chi(\lambda) = (\lambda - \lambda_1)(\lambda - \lambda_2) \cdots (\lambda - \lambda_n), where \lambda_1, \lambda_2, \cdots \lambda_n are the eigenvalues of A (with possible repetitions). Multiplying factors (\lambda - \lambda_1)(\lambda - \lambda_2) \cdots (\lambda - \lambda_n) and comparing the coefficients of \lambda^{n-1}, we deduce c_1 = -(\lambda_1 + \lambda_2 + \cdots + \lambda_n).

    Let N = {1, 2, n} and S_n denote the symmetric group of degree n (S_n = \left\{ \sigma \ | \ \sigma: N \rightarrow N, \sigma \ \text{bijective} \right\}). Let \iota be the identity element of S_n (\iota(1) = 1, \iota(2) = 2, \cdots \iota(n) = n).

    Using the Leibniz formula for determinants,

    |E| = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \ e_{1\sigma(1)}e_{2\sigma(2)} \cdots e_{n\sigma(n)} = e_{11}e_{22} \cdots e_{nn} \ + \ \sum_{\sigma \neq \iota} \text{sgn}(\sigma) \ e_{1\sigma(1)}e_{2\sigma(2)} \cdots e_{n\sigma(n)}

    For \sigma \neq \iota, consider an arbitrary term e_{1\sigma(1)}e_{2\sigma(2)} \cdots e_{n\sigma(n)}. There exists an i \in N such that \sigma(i) = j \ \text{with} \ j \neq i. Since \sigma(i) = j and \sigma is injective, \sigma(j) = k \ \text{with} \ k \neq j. Hence, e_{1\sigma(1)}e_{2\sigma(2)} \cdots e_{n\sigma(n)} contains at least two distinct non diagonal entries of E: e_{ij} and e_{jk}; therefore, n - 2 is the highest power of \lambda it can have.

    We deduce the product of Es diagonal terms e_{11}e_{22} \cdots e_{nn} = (\lambda - a_{11})(\lambda - a_{22}) \cdots (\lambda - a_{nn}) = \lambda^n + d_1\lambda^{n - 1} + \cdots + d_n includes all \chi(\lambda) \ \lambda^n \ \text{and} \ \lambda^{n - 1} terms.

    As shown previously, d_1 = -(a_{11} + a_{22} + \cdots + a_{nn}).

    Comparing \lambda^{n - 1} coefficients, we conclude:

    c_1 = d_1

    -(\lambda_1 + \lambda_2 + \cdots + \lambda_n) = -(a_{11} + a_{22} + \cdots + a_{nn})

    \sum_{i = 1}^n \lambda_i = tr(A)
    Last edited by kstahmer; June 3rd 2012 at 11:25 AM.
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