Linear Algebra-Matrix Similarity, Contract
Matrix similarity: P − 1 AP = B, denoted as A ∽ B, similarity transformation does not change the matrix eigenvalue P^{-1}AP = B, denoted as A∽B, similarity transformation does not change the matrix eigenvalueP−1AP=B , denoted as A∽B , Phase similar change change no change change moment matrix Laid syndrome value
Matrix contract: QTAQ = B, denoted as A ≃ B, contract transformation does not change the inertia index of the matrix Q^TAQ = B, denoted as A≃B, contract transformation does not change the inertia index of the matrixQTAQ=B,Marked as A≃B , together with the variable change does not change the variable moment matrix of inertia of the finger number
Similarity transformation steps:
find the eigenvalues→see the corresponding eigenvectors→get the transformation matrix P→ P − 1 AP = BP^{-1}AP = BP−1AP=B
Orthogonal similar transformation steps:
find the eigenvalues→see the corresponding eigenvectors→get the transformation matrix P→P orthogonal unity to Q→ QTAQ = BQ^{T}AQ = BQTAQ=B
Orthogonal unitization makes P become an orthogonal matrix Q, that is, QT = Q − 1, so this is still a similar transformation, and it is also a contract transformation. \color {blue} Orthogonal unitization makes P become an orthogonal matrix Q, that is, Q^T = Q^{-1}, so this is still a similar transformation, and it is also a contract transformation.N pay single bit of the so obtained P becomes as positive post rectangular matrix Q ,I.e. QT=Q−1,The order which also is a th phase similar change change ,The same sample also is a Species together with variable change .
The steps of contract transformation:
There is only one kind of similarity transformation, but there are many kinds of contract transformations, because it is enough to keep the inertia index unchanged. The common method isOrthogonal Similarity TransformationorMatching method。
Ideas to deal with the problem
When transforming the quadratic form into the standard form, there are two methods: 1. Orthogonal similar transformation, the standard form is unique 2. The matching method, the standard form is not unique; when the quadratic form is transformed into the standard form, similar transformation can be used to find Inertia exponent, write the canonical form directly, but you cannot write the transformation matrix at this time. If the transformation matrix is required, a matching method is required. Use y 1, y 2, y 3 to represent x 1, x 2, x 3 to obtain x = Q y y_1, y_2, y_3 represent x_1, x_2, x_3 and get x = QyY1,Y2,Y3Table shows x1,x2,x3Seeking to obtain x=Q y
For a special real symmetric matrix, its eigenvectors are orthogonal to each other, and the contract transformation matrix can be obtained only by unitization.