# L. (20 pulita total) Given is A-| 1 3 . Tins matrix has eigruvalunAi-Ag-2 and la-s....

###### Question:

l. (20 pulita total) Given is A-| 1 3 . Tins matrix has eigruvalunAi-Ag-2 and la-s. (a) (2 points) Explain wby the matrix A can be orthogonally dingoaleed (b) (4 points) Show that i is a l-for the eigenspace eurrespcoding to 0 Xi- "xz- -2 (c) (3 points) Show thatxis a basis for the eigeneqpace responding to A (d) (5 points) plc by an orthoyonal basis Hint: This can be dose by applying the Gra- by Schmidt process to (x. Xl to get (vi.V).Or, alternatively, by setting vx and replncing -x+bxa and determining e and b so that voLv (e) (3 points) Nonwalee the vectors vi. va, and va s to replace the orthogonal basis v VaVs by an orthosormal basisu u () (3 points) Use the veetors obtained in part (e) to make an orthogonal matrix P which allows one to orthogonally dingomalize A. Verify that D PTAP, where D is a diagonal matrix

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