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https://texercises.com/exercise/orthogonal-matrices/
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Exercise:
Ain M_ntimes nmathbbR textis orthogonal &iff A^TAI_N &iff AA^TI_n &iff A^-A^T. Ain M_ntimes nmathbbC textis unitary &iff A^*AI_Nquad A^*A^ToverlineAI &iff AA^*I_n &iff A^-A^T.

Solution:
Proof of second Lemma. Write A pmatrix vdots & & vdots v_ & hdots & v_n vdots & & vdots pmatrix v_kin mathbbC^n_textcol k...n. A^ToverlineA pmatrix hdots & v_^T & hdots & vdots & hdots & v_n^T & hdots pmatrix pmatrix vdots & & vdots overlinev_ & hdots & overlinev_n vdots & & vdots pmatrix v_i^T overlinev_j_leq ileq n leq jleq n langle v_iv_jrangle_leq ileq n leq jleq nquad * Now A is unitary iff v_...v_n are an orthogonal system iff *I_n. The equivalence of these to overlineAA^TI_n.
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Exercise:
Ain M_ntimes nmathbbR textis orthogonal &iff A^TAI_N &iff AA^TI_n &iff A^-A^T. Ain M_ntimes nmathbbC textis unitary &iff A^*AI_Nquad A^*A^ToverlineAI &iff AA^*I_n &iff A^-A^T.

Solution:
Proof of second Lemma. Write A pmatrix vdots & & vdots v_ & hdots & v_n vdots & & vdots pmatrix v_kin mathbbC^n_textcol k...n. A^ToverlineA pmatrix hdots & v_^T & hdots & vdots & hdots & v_n^T & hdots pmatrix pmatrix vdots & & vdots overlinev_ & hdots & overlinev_n vdots & & vdots pmatrix v_i^T overlinev_j_leq ileq n leq jleq n langle v_iv_jrangle_leq ileq n leq jleq nquad * Now A is unitary iff v_...v_n are an orthogonal system iff *I_n. The equivalence of these to overlineAA^TI_n.
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eth, fs23, lineare algebra, matrices, orthogonal, proof
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