2 Decomposition of the matrix

2. Decomposition of matrices

Matrix decomposition (decomposition, factorization) is the process of splitting a matrix into the product or addition of several matrices, which can be divided into triangular decomposition, full rank decomposition, QR decomposition, Jordan decomposition, SVD (singular value) decomposition, and spectral decomposition, in which triangular decomposition (LU decomposition) is another form of Gaussian elimination, in the linear algebra of the undergraduate has been used to rot, Jordan decomposition in the previous chapter on the introduction of linear algebra "for Jordan standard form" has been introduced. This chapter only introduces QR decomposition, full rank decomposition, SVD (singular value) decomposition, and spectral decomposition. 2.1. QR Decomposition

Describe
A=qr
A is the full rank matrix, q is the orthogonal matrices, R is the upper triangular array, decomposition unique
A=ur (same as for converting orthogonal matrices to unitary matrices)
If A is just a column full rank, (Amxn,n≤m a_{mxn}, N≤m, rank n) Then
Amxn=qmxnrnxn A_{MXN} = q_{mxn}r_{nxn}, Q as long as the N-column vector standard orthogonal can be met, R or the upper triangular array

The QR Decomposition step asks R (a) to determine if a is full rank by column a= (X1,X2,X3) a= (x_1, X_2, x_3), orthogonal to Y1,y2,y3 y_1, y_2,y_3, Unit z1,z2,z3 z_1, z_2, z_3 order
Q= (Z1,Z2,Z3) q= (z_1,z_2,z_3)
r=⎛⎝⎜| | y1| | XX (X2,Z1) | | y2| | 0 (X3,Z1) (x3