notes3 - Stat 430/Math468 Notes #3 Matrix Algebra and...

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Stat 430/Math468 – Notes #3 Chapter 2 Matrix Algebra and Random Vectors (continued) Orthogonal Matrix The square matrix Q is called an orthogonal matrix if '' QQ QQ I = = , where I is the identity matrix of the same dimension. Remark: 1. From the definition, . 1 ' QQ = 2. All column vectors in Q have unit length and are mutually perpendicular (orthogonal). The row vectors of Q have the same property. Example: 11 10 22 , 01 1 1 ⎛⎞ ⎜⎟ ⎝⎠ Trace of a Matrix Let be a 11 12 1 21 22 2 12 ... ... () :: : : ... k k ij kk k k aa a a Aa a == × matrix. The trace of is . A 1 k ii i tr A a = = Results about Trace Let , A B be two × matrices and c be a scalar. Then 1) ( ) tr cA ctr A = 2) ( ) tr A B tr A tr B ±= ± 3) ( tr AB tr BA = ) ) 4) 1 ( tr B AB tr A = 5) 2 (') ij ij tr A A a = ∑∑ Eigenvalue and Eigenvector 1. Let be a matrix and A × I be a × identity matrix. Then the scalars , ,..., k λ satisfying the polynomial equation (characteristic equation) ()| |0 fI A =− = are called the eigenvalues (characteristic roots) of A .
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notes3 - Stat 430/Math468 Notes #3 Matrix Algebra and...

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