8) Diagonalization and the Spectral Theorem

Question 1

What is a similar matrix

Answer 1

A matrix A ∈ Mn(K) is similar to B ∈ Mn(K) if there is an invertible
matrix P ∈ Mn(K) such that B = P AP^ −1

Question 2

What properties do similar matrices share

Answer 2

• Rank
• Trace
• Determinant
• Eigenvalues

Question 3

How can we calculate high powers of a matrix using diagonable matrices

Answer 3

Question 4

In regards to the basis and eignevalues of a matrix, when is a matrix diagonalizable

Answer 4

Question 5

In regards to the algebraic and geometric multiplicitices of a matrix, when is a matrix diagonalizable

Answer 5

The matrix A ∈ Mn(K) is diagonalizable if and only if the algebraic
and geometric multiplicities of each of its eigenvalues coincide

Question 6

When is a matrix orthogonal

Answer 6

A matrix P ∈ Mn(R) is orthogonal if and only if its columns form an orthonormal basis for R^n

Question 6

What does it mean for two matrices to be orthogonally similar

Answer 6

Let A, B ∈ Mn(R). If there exists an orthogonal matrix P ∈ Mn(R) such that P^TAP = B, then we say that A and B are orthogonally similar

Question 7

When is a matrix orthogonally diagonalizable

Answer 7

If it is orthogonally similar to a diagonal matrix

Question 8

Is an orthogonally diagonalizable matrix symmetric

Answer 8

If A ∈ Mn(R) is orthogonally diagonalizable then A is symmetric

Question 9

Describe the proof that an orthogonally diagonalizable matrix is symmetric

Answer 9

Question 10

Whats is a unitary matrix

Answer 10

If the columns of a matrix U ∈ Mn(C) form an orthonormal basis for
Cn , then U is called unitary.
(This is the complex analogue of an orthogonal matrix)

Question 11

What does it mean for two matrices to be unitarily similar

Answer 11

If there exists a unitary matrix U such that U^∗AU = B, then we say that A and B are unitarily similar

Question 12

What does it mean for a matrix to be Hermitian

Answer 12

A matrix A ∈ Mn(C) is Hermitian if A^∗ = A.

Question 13

What does it mean about the eigenvalues of a matrix if it is a Hermitian matrix

Answer 13

Then all the eigenvalues of A are real

Question 14

If A is a Hermitian matrix, then if λ1 and λ2 are distinct eigenvalues
with corresponding eigenvectors v1 and v2 respectively what can we say about v1 and v2

Answer 14

Then v1 and v2 are orthogonal

Question 15

What does it mean for a matrix to be normal

Answer 15

Question 16

What is Schur’s Theorem

Answer 16

If A ∈ Mn(C), then A is unitarily similar to an upper triangular matrix T. Moreover, the diagonal entries of T are the eigenvalues of A.

Question 17

What is Schur decomposition of a matrix

Answer 17

A = UT U^∗

Question 18

What is The Spectral Theorem for Hermitian matrices

Answer 18

Question 19

What is a spectral decomposition of a amtrix

Answer 19

Question 20

What is a Singular Value Decomposition

Answer 20