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Chapter 5: Q5.4-26E (page 267)

It can be shown that the trace of a matrix \(A\) equals the sum of the eigenvalues of \(A\). Verify this statement for the case \(A\) is diagonalizable.

Short Answer

Expert verified

It is proved that trace of \(A\) is equal to sum of its three eigenvalues, even when it is diagonalizable.

Step by step solution

01

Use the given information

Let \(A\) be a diagonalizable matrix such that it can be represented in the form \(PD{P^{ - 1}}\) , where \(P\) is an invertible matrix and \(D\) is the diagonal matrix. So, we can write as follows:

\(A = PD{P^{ - 1}}\)

02

Prove the given statement

As \(P\) is an invertible matrix, It can be written that

\(\begin{aligned}{}{\rm{tr}}\,A &= {\rm{tr}}\,\left( {\left( {PD} \right){P^{ - 1}}} \right)\\ &= {\rm{tr}}\,\left( {{P^{ - 1}}PD} \right)\\ &= {\rm{tr}}\,D\end{aligned}\)

Matrix \(D\) is a diagonal matrix which consist 3 eigenvalues on its main diagonal. So, \({\rm{tr}}\,A = {\rm{tr}}\,D\) represents that the trace of \(D\), that is, sum of three eigenvalues of \(D\) is equal to trace of \(A\).

Thus, trace of \(A\) is equal to sum of its three eigenvalues, even when it is diagonalizable.

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Most popular questions from this chapter

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

  1. If \(A\) is \(3 \times 3\), with columns \({{\rm{a}}_1}\), \({{\rm{a}}_2}\), and \({{\rm{a}}_3}\), then \(\det A\) equals the volume of the parallelepiped determined by \({{\rm{a}}_1}\), \({{\rm{a}}_2}\), and \({{\rm{a}}_3}\).
  2. \(\det {A^T} = \left( { - 1} \right)\det A\).
  3. The multiplicity of a root \(r\) of the characteristic equation of \(A\) is called the algebraic multiplicity of \(r\) as an eigenvalue of \(A\).
  4. A row replacement operation on \(A\) does not change the eigenvalues.

Question 18: It can be shown that the algebraic multiplicity of an eigenvalue \(\lambda \) is always greater than or equal to the dimension of the eigenspace corresponding to \(\lambda \). Find \(h\) in the matrix \(A\) below such that the eigenspace for \(\lambda = 5\) is two-dimensional:

\[A = \left[ {\begin{array}{*{20}{c}}5&{ - 2}&6&{ - 1}\\0&3&h&0\\0&0&5&4\\0&0&0&1\end{array}} \right]\]

Consider an invertible n 脳 n matrix A such that the zero state is a stable equilibrium of the dynamical system x鈫�(t+1)=ATx鈫�(t)What can you say about the stability of the systems.

x鈫�(t+1)=ATx鈫�(t)

Question: Is \(\left( {\begin{array}{*{20}{c}}4\\{ - 3}\\1\end{array}} \right)\) an eigenvector of \(\left( {\begin{array}{*{20}{c}}3&7&9\\{ - 4}&{ - 5}&1\\2&4&4\end{array}} \right)\)? If so, find the eigenvalue.

Question: In Exercises \({\bf{3}}\) and \({\bf{4}}\), use the factorization \(A = PD{P^{ - {\bf{1}}}}\) to compute \({A^k}\) where \(k\) represents an arbitrary positive integer.

3. \(\left( {\begin{array}{*{20}{c}}a&0\\{3\left( {a - b} \right)}&b\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0\\3&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}a&0\\0&b\end{array}} \right)\left( {\begin{array}{*{20}{c}}1&0\\{ - 3}&1\end{array}} \right)\)
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