Questions for repetition: diagonalization by orthogonal matrices
-
Describe the steps leading to a full definition of the set of complex numbers (imaginary unit, complex numbers, operations with them, absolute value, conjugate number).
-
What are matrix analogs of a) real numbers, b) conjugation and c) an absolute value of a complex number (without proof)?
-
Write out and prove the properties of the scalar product in
.
-
Assuming the Euler formula known, derive the polar form of a complex number. What can you say about the angle
in the polar form?
-
Prove that a quadratic form of a matrix is homogeneous of degree 2.
-
Prove that a quadratic form takes values in the set of numbers.
-
Define positive and non-negative matrices. State Sylvester's criterion.
-
Illustrate the Sylvester criterion geometrically.
-
Show that
is symmetric and non-negative.
-
Give the definition of a basis and derive the formula for the coefficients
in the decomposition
for an arbitrary vector
.
-
How are the decomposition coefficients transformed when one passes from one basis to another?
-
Give the full picture behind the similarity definition.
-
Prove that for an orthogonal matrix, a) the inverse and transpose are the same, b) the transpose and inverse are orthogonal.
-
An orthogonal matrix preserves scalar products, norms and angles.
-
If you put elements of an orthonormal basis side by side, the resulting matrix will be orthogonal.
-
The transition matrix from one orthonormal basis to another is orthogonal.
-
Show that the product of two diagonal matrices is diagonal.
-
Define eigenvalues and eigenvectors. Why are we interested in them?
-
What is the link between eigenvalues and a characteristic equation of a matrix?
-
Prove that the characteristic equation is a polynomial of order
, if
is of size
.
-
Prove that in
any matrix has at least one eigenvector.
-
A symmetric matrix in
has only real eigenvalues.
-
If
is symmetric, then it has at least one real eigenvector.
-
A symmetric matrix has at least one eigenvector in any nontrivial invariant subspace.
-
What is the relationship between the spectra of
and
?
-
A matrix diagonalizable by an orthogonal matrix must be symmetric.
-
For a symmetric matrix, an orthogonal complement of an invariant subspace is invariant.
-
Main theorem. A symmetric matrix is diagonalizable by an orthogonal matrix.
Leave a Reply
You must be logged in to post a comment.