hw06 - MTH 201 Problem Set#6 Due Wednesday May 19...

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MTH 201 Problem Set #6 Due: Wednesday, May 19, 2:15pm (Self-Adjoint Operators) 1. Show that the induced norm (i.e., || || , x xx = ) satisfies all the axioms of norm. 2. Consider the following matrix. 01 0 A1 0 i ii i + ⎛⎞ ⎜⎟ =− + ⎝⎠ 1) Show that A is self-adjoint under the complex Euclidean inner product. 2) Find the eigenvalues and eigenvectors of A. 3) Show that eigenvectors are mutually orthogonal. 3. Consider the following three-dimensional stress condition at a point in a structure. x σ =140 MPa, xy τ =0 MPa, xz =60 MPa, y = 100 MPa, yz =0 MPa, z =140 MPa Then, the stress condition is described as a symmetric matrix to satisfy force equilibrium as follows: 0 140 60 0 60 140 0 100 0 A 1) Find the principal stresses and the orientation of principal axes of stress by solving the eigenvalue problem. (Hint: principal stresses are eigenvalues of A, and principle axes are corresponding eigenvectors.) 2) The material used for the structure is known to fail at the tensile stress of 190 MPa. Will this structure fail under the given stress condition? (Hint: Compare the principle stresses with the material strength (190
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