W8 W9 Linear Algebra OSU

What is an eigenvector?

a nonzero vector x such that Ax = λx for some scalar λ

What is an eigenvalue?

A scalar λ such that there exists a vector x so Ax = λx

How do you determine if λ is an eigenvalue of A?

Check if (A-λI)x = 0 has a nontrivial solution

What is the eigenspace E_λ​?

The set of all solutions to (A−λI)x=0

it's a subspace of R^n

What are the eigenvalues of a triangular matrix?

The entries on its main diagonal.

Are eigenvectors with different eigenvalues linearly independent?

Yes, always

What is the characteristic equation of a matrix A?

det(A-xI) = 0

Can a matrix with real entries have complex eigenvalues?

Yes

What does it mean if the rref of (A-λI |0) has only one trivial solution?

λ is not an eigenvalue of A

What do you get when you solve det(A - xI) = 0?

The eigenvalues of A

What does it mean if two nxn matrices are similar?

there exists an invertible matrix P

A = PBP^-1

B is just another matrix representing the same linear transformation but w respect to a different basis

Do similar matrices have the same eigenvalues?

Yes, similar matrices have the same eigenvalue

Is the converse true — do matrices with the same eigenvalues have to be similar?

No, having the same eigenvalues does not guarantee similarity

What is the algebraic multiplicity of an eigenvalue?

The algebraic multiplicity is the highest power of (x - λ) dividing the characteristic polynomial det⁡(A−xI)

What is the geometric multiplicity of an eigenvalue?

The geometric multiplicity is the dimension of the eigenspace

E_λ = Null(A-λI)

How do algebraic and geometric multiplicities relate?

Geometric multiplicity ≤ algebraic multiplicity

What is a defective matrix?

A matrix is defective if for some eigenvalue, the geometric multiplicity is less than the algebraic multiplicity

When is a matrix diagonalizable?

If it has n linearly independent eigenvectors, or equivalently, if it is similar to a diagonal matrix

How do you diagonalize a matrix A?

Find the invertible P whose columns are eigenvectors and diagonal D of eigenvalues so that A = PDP^-1

What is the diagonal matrix D in the diagonalization of a matrix A?

D is a diagonal matrix whose diagonal entries are the eigenvalues of A

What happens if a matrix has nnn distinct eigenvalues?

It is guaranteed to be diagonalizable

What is the matrix of a linear transformation T with respect to basis B?

It is the matrix whose columns are the coordinate vectors

[T(bi)]_B

What is the importance of the basis formed by eigenvectors?

It diagonalizes the linear transformation, simplifying computations like powers of the matrix

What is the complex conjugate of a complex number z = a + bi?

The complex conjugate is zˉ = a + bi

If λ= = a + bi is an eigenvalue of a real matrix A, what other eigenvalue must A also have?

The conjugate eigenvalue λˉ =a−bi

Do eigenvalues of matrices with complex entries always come in conjugate pairs?

No, this is only true for matrices with real entries

What property defines a symmetric matrix A?

A is symmetric if ^T = A

What type of eigenvalues do real symmetric matrices have?

Real symmetric matrices have real eigenvalues.

How does the conjugate transpose operation apply to matrices?

By taking the complex conjugate of each entry (and transposing if applicable).

What is a key difference in eigenvalues between matrices with real entries vs. matrices with complex entries?

Real matrices have complex eigenvalues in conjugate pairs; complex-entry matrices do not necessarily have this property.