Linear

Linear Algebra and Matrix Analysis for Statistics by Sudipto Banerjee

By Sudipto Banerjee

"Linear algebra and the examine of matrix algorithms became basic to the advance of statistical types. utilizing a vector-space strategy, this ebook offers an knowing of the foremost thoughts that underlie linear algebra and matrix research. each one bankruptcy introduces a key subject, corresponding to infinite-dimensional areas, and offers illustrative examples. The authors research fresh advancements in diverse Read more...

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Linear algebra and the research of matrix algorithms became basic to the advance of statistical types. utilizing a vector house procedure, this booklet offers an knowing of the major Read more...

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Where j ∈ {m1 + 1, . . , m}. These rows become the i-th and j-th columns of A as below:   ··· ··· ··· ··· ··· ···  ai1 · · · ain1 ain1 +1 · · · ain     ··· ··· ··· ··· ··· ···     and A=    ··· ···  · · · · · · · · · · · ·    aj1 · · · ajn1 ajn1 +1 · · · ajn  ··· ··· ··· ··· ··· ···   .. .. . a . a . i1 j1   .. .. ..     . . .     .. ..   . a . a . in1 jn1    . A =    .  . . a ..   .. ain +1 .. jn1 +1 1    . .. ..   .. . .    ..

U1 vn  u2   u2 v1 u2 v2 . . u2 vn      uv =  . [v1 , v2 , . . 8)  . .. ..  .    . . um um v1 um v2 . . um vn This product of two vectors is often called the outer product. In short, we write uv = [ui vj ]m,n i,j=1 . We point out that the outer product is defined for two vectors of any dimensions, and produces a matrix as a result. This is in contrast with the inner product or inner product, which is defined only for two vectors of the same dimension and produces a scalar.

Put differently, the bandwidth is the smallest number of adjacent diagonals to which the nonzero elements are confined. 28 MATRICES, VECTORS AND THEIR OPERATIONS Band matrices need not be square matrices. A 6×5 band matrix with lower bandwidth p1 = 1 and upper bandwidth p2 = 2 has the following structure:   ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ 0   0 ∗ ∗ ∗ ∗   0 0 ∗ ∗ ∗ .   0 0 0 ∗ ∗ 0 0 0 0 ∗ The bandwidth is p1 + p2 + 1 = 4, which can be regarded as the length of the “band” beyond which all entries are zero.

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