Let T and U be a self-adjoint linear operators on an n-dimensional inner product space V, and let A = [T]3, where 3 is an orthonormal basis for V. Prove the following results. (a) T is positive definite [semidefinite] if and only if all of its eigenval- ues are positive [nonnegative]. (b) T is positive definite if and only if Σ Aijajāį > 0 for all nonzero n-tuples (a₁, a2, ..., an). i.i

Let T and U be a self-adjoint linear operators on an n-dimensional inner product space V, and let A = [T]3, where 3 is an orthonormal basis for V. Prove the following results. (a) T is positive definite [semidefinite] if and only if all of its eigenval- ues are positive [nonnegative]. (b) T is positive definite if and only if Σ Aijajāį > 0 for all nonzero n-tuples (a₁, a2, ..., an). i.i

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.3: Orthonormal Bases:gram-schmidt Process

Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...

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