Unit 3 Linear Algebra

What is an eigenvector of A?

A nonzero vector v satisfying Av = λv for some scalar λ.

What is an eigenvalue of A?

A scalar λ for which Av = λv has a nonzero solution.

Can the zero vector be an eigenvector?

No. Eigenvectors must be nonzero.

Can zero be an eigenvalue?

Yes. Zero is an eigenvalue if Av = 0 has a nonzero solution.

How do you verify that v is an eigenvector?

Compute Av and check whether it equals one scalar multiple λv.

What equation finds eigenvalues?

det(A - λI) = 0.

What is the characteristic polynomial?

p(λ) = det(A - λI).

What are the eigenvalues of a triangular matrix?

The entries on its main diagonal.

Why does det(A - λI) = 0 identify eigenvalues?

It makes A - λI singular, so (A - λI)x = 0 has a nonzero solution.

How do you find the eigenspace for λ?

Solve (A - λI)x = 0.

What is the eigenspace formula?

Eλ = Nul(A - λI).

What is the difference between an eigenvector and an eigenspace?

An eigenvector is one nonzero vector; an eigenspace is the entire subspace of solutions, including zero.

Does an eigenspace contain zero?

Yes, because every subspace contains zero.

How do you find a basis for an eigenspace?

Row-reduce A - λI and use the special solution vectors.

What is algebraic multiplicity?

The number of times an eigenvalue is repeated as a root of the characteristic polynomial.

What is geometric multiplicity?

The dimension of the eigenspace.

What inequality relates the two multiplicities?

1 ≤ geometric multiplicity ≤ algebraic multiplicity.

What does it mean for A to be diagonalizable?

A = PDP^(-1), where P is invertible and D is diagonal.

What are the columns of P?

Linearly independent eigenvectors of A.

What are the diagonal entries of D?

The eigenvalues matching the eigenvectors in the corresponding columns of P.

When is an n by n matrix diagonalizable?

When it has n linearly independent eigenvectors.

If A has n distinct eigenvalues, is it diagonalizable?

Yes.

Can a matrix with repeated eigenvalues be diagonalizable?

Yes, if it still has enough linearly independent eigenvectors.

Is every diagonalizable matrix invertible?

No. It may have zero as an eigenvalue.

When is a diagonalizable matrix invertible?

When none of its eigenvalues are zero.

How do you compute A^k using diagonalization?

A^k = PD^kP^(-1).

How do you compute D^k?

Raise each diagonal entry to the kth power.

If Av = λv, what is A^kv?

A^kv = λ^kv.

What does an eigenvector represent geometrically?

A direction that is not rotated away from its line by the transformation.

What does a positive eigenvalue do?

Scales the eigenvector without reversing its direction.

What does a negative eigenvalue do?

Scales and reverses the eigenvector direction.

What does λ = 1 mean?

The eigenvector is unchanged.

What does λ = -1 mean?

The eigenvector is reversed with the same length.

What does λ = 0 mean?

The eigenvector is mapped to zero.

What does |λ| > 1 mean?

The transformation stretches along that eigenvector direction.

What does 0 < |λ| < 1 mean?

The transformation shrinks along that eigenvector direction.

What does it mean for two matrices to be similar?

A = PBP^(-1) for some invertible P.

What do similar matrices share?

The same eigenvalues, characteristic polynomial, determinant, and trace.

What MATLAB command finds eigenvalues and eigenvectors?

[V,D] = eig(A)

What does V contain in [V,D] = eig(A)?

Eigenvectors as columns.

What does D contain in [V,D] = eig(A)?

Eigenvalues on the diagonal.

What MATLAB command finds only eigenvalues?

eig(A)

How can MATLAB find the eigenspace for λ?

null(A - lambda*eye(size(A)))

What MATLAB command computes matrix powers?

A^k

What is a stochastic matrix?

A nonnegative matrix whose columns sum to 1 under the column-vector convention.

What is the Markov-chain update formula?

x_(k+1) = Px_k.

How do you compute x1?

x1 = Px0.

How do you compute x2?

x2 = Px1 = P^2x0.

What is the general Markov-chain formula?

x_k = P^kx0.

How do you build a transition matrix from words?

Each column represents the current state, and each row represents the next state.

What should each column of a stochastic matrix sum to?

1.

What is a steady-state vector?

A vector q that satisfies Pq = q.

What equation is used to solve for steady state?

(P - I)q = 0.

What additional equation is needed for a probability steady state?

The entries of q must sum to 1.

What additional condition is needed for a population steady state?

The entries must sum to the total population.

Which eigenvalue corresponds to steady state?

λ = 1.

Is a steady-state vector an eigenvector?

Yes, an eigenvector associated with λ = 1.

How can MATLAB approximate a long-term Markov state?

P^100*x0

What is a linear transformation?

A function preserving vector addition and scalar multiplication.

What are the two linearity conditions?

T(u + v) = T(u) + T(v) and T(cu) = cT(u).

Is every transformation T(x) = Ax linear?

Yes.

What is the standard matrix of T from Rn to Rm?

A = [T(e1) T(e2) … T(en)].

What is the most important matrix-transformation rule?

Apply T to each domain basis vector and place the output coordinate vectors as columns.

What should you identify first in a transformation problem?

The domain, codomain, and ordered bases.

What should you do second in a transformation problem?

Apply T to each domain basis vector.

What should you do third in a transformation problem?

Express each output in the codomain basis.

What should you do fourth in a transformation problem?

Place those coordinate vectors into the columns of the matrix.

How many rows does a transformation matrix have?

The dimension of the codomain.

How many columns does a transformation matrix have?

The dimension of the domain.

How do you find the standard matrix for T:R2→R2?

Compute T(e1) and T(e2) and use them as the columns.

How do you find a matrix relative to a nonstandard basis B?

Compute T(b1), T(b2), and so on, then express each result in B-coordinates.

What is the basis matrix P?

The matrix whose columns are the basis vectors in standard coordinates.

When the same basis B is used in domain and codomain, how do you find the standard matrix?

A = P[T]B P^(-1).

How do you show two vectors form a basis for R2?

Show they are linearly independent, often by checking the determinant is nonzero.

How do you handle polynomial transformations?

Apply T to 1, t, t^2, and other basis polynomials, then use output coefficients as columns.

What is the coordinate vector of a + bt + ct^2 relative to {1,t,t^2}?

[a;b;c]

For T:P2→P4, what size is the matrix?

5 by 3.

Why is a P2→P4 matrix 5 by 3?

P2 has dimension 3 and P4 has dimension 5.

What is the dot-product formula?

u·v = u1v1 + u2v2 + … + unvn.

What is the vector-length formula?

||v|| = sqrt(v·v).

What is the distance formula?

dist(u,v) = ||u - v||.

What is the angle formula?

cos(theta) = (u·v)/(||u||||v||).

What MATLAB command computes a dot product?

dot(u,v)

What MATLAB command computes vector length?

norm(v)

What MATLAB command computes distance?

norm(u-v)

What MATLAB command computes an angle in degrees?

acosd(dot(u,v)/(norm(u)*norm(v)))

When are two vectors orthogonal?

When their dot product is zero.

What is an orthogonal set?

A set where every pair of different vectors has dot product zero.

What is an orthonormal set?

An orthogonal set where every vector also has length 1.

How do you normalize a vector?

v/||v||

What equation shows that Q has orthonormal columns?

Q^TQ = I.

If Q is square with orthonormal columns, what is Q^(-1)?

Q^(-1) = Q^T.

Are nonzero orthogonal vectors linearly independent?

Yes.

What is the projection of y onto u?

proj_u(y) = ((y·u)/(u·u))u.

What is the projection formula when u is a unit vector?

proj_u(y) = (y·u)u.

What is projection onto W with an orthogonal basis {u1,…,up}?

proj_W(y) = Σ((y·ui)/(ui·ui))ui.

What is the closest vector in W to y?

proj_W(y).

What is the purpose of Gram-Schmidt?

To turn a linearly independent basis into an orthogonal basis for the same subspace.

What is the first Gram-Schmidt step?

u1 = v1.

What is the second Gram-Schmidt step?

u2 = v2 - proj_u1(v2).