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Spanning matrix space by niltopent matrices: TIFR 2018 Part A, Problem 15

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" hover_enabled="0" _i="1" _address="0.0.0.1"]The set of nilpotent matrices in \(M_3(R)\) spans \(M_3(R)\) considered as an R-vector space (a matrix A is said to be nilpotent if there exists n ∈ N such that \(A^n = 0)\). [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" _i="0" _address="0.1.0.0.0" hover_enabled="0"]TIFR 2018 Part A, Problem 15 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" _i="1" _address="0.1.0.0.1" open="off" hover_enabled="0"]LINEAR  ALGEBRA [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" _i="2" _address="0.1.0.0.2" open="off" hover_enabled="0"]Medium [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" _i="3" _address="0.1.0.0.3" open="off" hover_enabled="0"]Linear Algebra, Hoffman and Kunze [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

Start with hints

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Let’s first try an easier problem first :
  • Whether there exists a and b such that I = a N + b M,where N and M are nilpotent matrices and a,b are real numbers ?
  • To use the given definition to prove or disprove this we need to take powers and things will become too messy for larger n though here it is 3.[We have to take only \(A^3=0\) it is a small exercise to verify]
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  • So let us recapitulate some properties of nilpotent matrices: They are the matrices which has only eigenvalue 0. How to use this?
  • We wish to have some linear operator acting on the right side for the RHS is a linear combination and the natural choice is Trace.
  • Trace of a nilpotent matrix is 0.
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  • Now assume that I is a finite linear combination of Nilpotent matrices and then take Trace Operator on both sides.
  • Thus LHS has trace n and RHS has trace 0.(Contradiction)
  • So the answer is False.
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Food for Thought:
  • Observe that the order of the matrices is not at all a big issue here!
  • Does span of the nilpotent matrices contain all the matrices with trace 0?
  • Hint:Try to find all the 2x2 nilpotent matrices and check the above statement out!
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Watch the video

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Similar Problems

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