HP 48gII - Manual (Page 376)

Page 11-50
The Singular Value Decomposition (SVD) of a rectangular matrix
A
m × n
consists
in determining the matrices U, S, and V, such that A
m × n
= U
m × m
⋅S
m × n
⋅V T
n × n
,
where U and V are orthogonal matrices, and S is a diagonal matrix. The
diagonal elements of S are called the singular values of A and are usually
ordered so that s
i
≥ s
i+1
, for i = 1, 2, …, n-1. The columns [ u
j
] of U and [v
j
] of
V are the corresponding singular vectors.
Function SVD
In RPN, function SVD (Singular Value Decomposition) takes as input a matrix
A
n × m
, and returns the matrices U
n × n
, V
m × m
, and a vector s in stack levels 3, 2,
and 1, respectively. The dimension of vector s is equal to the minimum of the
values n and m. The matrices U and V are as defined earlier for singular
value decomposition, while the vector s represents the main diagonal of the
matrix S used earlier.
For example, in RPN mode:
[[5,4,-1],[2,-3,5],[7,2,8]] SVD
3: [[-0.27 0.81 –0.53][-0.37 –0.59 –0.72][-0.89 3.09E-3 0.46]]
2: [[ -0.68 –0.14 –0.72][ 0.42 0.73 –0.54][-0.60 0.67 0.44]]
1: [ 12.15 6.88 1.42]
Function SVL
Function SVL (Singular VaLues) returns the singular values of a matrix
A
n × m
as
a vector s whose dimension is equal to the minimum of the values n and m.
For example, in RPN mode, [[5,4,-1],[2,-3,5],[7,2,8]] SVL
produces [ 12.15 6.88 1.42].
Function SCHUR
In RPN mode, function SCHUR produces the Schur decomposition of a square
matrix
A returning matrices Q and T, in stack levels 2 and 1, respectively,
such that A = Q⋅T⋅Q T , where Q is an orthogonal matrix, and T is a triangular
matrix. For example, in RPN mode,
[[2,3,-1][5,4,-2][7,5,4]] SCHUR
results in:
2: [[0.66 –0.29 –0.70][-0.73 –0.01 –0.68][ -0.19 –0.96 0.21]]