Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

True, false questions about matrices

  1. Apr 7, 2009 #1
    Hello,
    could you help me with some true false questions about matrices?

    First
    If [tex]\mathbf{A}=\mathbf{B}-\mathrm{i}\mathbf{C}[/tex] is hermitian matrix [tex]\mathbf{B},\mathbf{C}[/tex] are real, then [tex]\mathbf{B},\mathbf{C}[/tex] are anti-symmetric matrices. True? False?

    My solution
    If [tex]\mathbf{A}[/tex] is hermitian, then [tex]\mathbf{A}^{\mathrm{H}}=\mathbf{A}[/tex] so [tex](\mathbf{B}-\mathrm{i}\mathbf{C})^{\mathrm{H}}=\mathbf{B}^{\mathrm{H}}-(\mathrm{i}\mathbf{C})^{\mathrm{H}}=\mathbf{B}^{\mathrm{T}}+i\mathbf{C}^{\mathrm{T}}[/tex]. It implies the fact [tex]\mathbf{B}^{\mathrm{T}}+i\mathbf{C}^{\mathrm{T}}=\mathbf{B}-\mathrm{i}\mathbf{C}[/tex]. [tex]\mathbf{B}[/tex] is symmetric and [tex]\mathbf{C}[/tex] is anti-symmetric. FALSE

    Second
    If A is diagonalizable and its eigenvalues are [tex]\{\lambda_1,\lambda_2,\cdots,\lambda_n\}[/tex], then [tex]\prod_{k=1}^{n}(x-\lambda_k)=0[/tex] has n different solutions

    My solution
    Matrix is diagonalizable, then [tex]\lambda_i\ne\lambda_j[/tex] for [tex]i\ne j[/tex]. So polynomial has n different soln's. TRUE


    Thank you very much for your help...
     
  2. jcsd
  3. Apr 7, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You have a much bigger problem that whether this statement is true of false! What makes you think that if a matrix is diagonalizable, then [itex]\lambda_i\ne\lambda_j[/itex]?

    The identity matrix is diagonalizable because it is diagonal. What are its eigenvalues?

    (An n by n matrix is diagonalizable if and only if it has n independent eigenvectors. It doesn't matter what the eigenvalues are.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: True, false questions about matrices
  1. True of false? (Replies: 5)

Loading...