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Things to know - Chapters 3, 5, 6 & 7
MA2321, Fall '03, T. Olson

Know these terms and relationships between them:

determinant
triangular matrix
diagonal matrix
transpose
inverse
eigenvalue
eigenvector
eigenspace
characteristic polynomial
characteristic equation
multiplicity
similar matrices
diagonalizable, diagonalization
dot product, inner product
orthogonal, perpendicular (vectors)
length, norm (vector)
distance (between vectors)
row space, column space, nullspace
orthogonal complement
orthogonal matrix
orthogonally diagonalizable
symmetric matrix
quadratic form

Things to think about:

1.: How does each elementary row operation affect the determinant of a matrix?
2.: How do you compute determinants by hand for a $2\times 2$ matrix? ...for a triangular matrix? ...using row reduction?
3.: What's the geometrical meaning of a determinant (in 2-D or 3-D)?
4.: How is the determinant of a matrix A related to the determinant of A^T? ...of A^-1? ...of AB?
5.: What does it say about A if $\det A=0$ ?
6.: What is an eigenvector?
7.: For a given matrix A, how do you decide if a given number is an eigenvalue of A?
8.: For a given matrix A, how do you decide if a given vector is an eigenvector of A? How do you find the corresponding eigenvalue?
9.: Can zero be an eigenvalue? Can the zero vector be an eigenvector?
10.: Given a matrix A and one eigenvalue of A, how do you compute the eigenspace associated with that eigenvalue?
11.: How does row reduction affect eigenvalues?
12.: What does it mean if zero is an eigenvalue of a matrix?
13.: What does it mean if two matrices are similar? What can you say about the eigenvalues, eigenvectors, and/or characteristic equations of two similar matrices?
14.: How is the characteristic polynomial related to the eigenvalues?
15.: How do you diagonalize a matrix (find the ``P'' and ``D'' in the factorization A=P D P^-1)? Under what conditions can you do this? When can you do this with an orthogonal P?
16.: How do you tell if two vectors are orthogonal? ...if a set of vectors is orthogonal?
17.: What is meant by ``the orthogonal complement of W''? If $W=\mbox{Span} \{ \vec{v}_1, \vec{v}_2, \ldots , \vec{v}_k \}$ , how can you tell if a given vector is in $W^{\perp}$ ?
18.: What is an ``orthogonal matrix''? What is its inverse?
19.: Suppose a non-square matrix U has orthonormal columns. What do you know about U^T U? ...about U U^T?
20.: Suppose a square matrix U has orthonormal columns. What do you know about U^T U? ...about U U^T?
21.: What properties are conserved under transformation by a matrix with orthonormal columns?
22.: How do you compute the projection of an arbitrary vector onto a subspace $W=\mbox{Span} \{ \vec{v}_1, \vec{v}_2, \ldots , \vec{v}_k \}$ ? (Give at least two answers, one assuming that the $\vec{v}_i$ s are orthonormal, and one assuming that they are only orthogonal).
23.: What is meant by ``the vector in $\mbox{Span} \{ \vec{v}_1, \vec{v}_2, \ldots , \vec{v}_k \}$ which is closest to y''? How would you compute it? How would you check your answer using a dot product?
24.: How can you split a given vector into the sum of two pieces: one vector in W and one vector in $W^{\perp}$ ?
25.: Given an orthogonal basis for $\mbox{I\hspace{-.14em}R}^2$ (or $\mbox{I\hspace{-.14em}R}^3$ ) and an arbitrary vector $\vec{y}$ in $\mbox{I\hspace{-.14em}R}^2$ (or $\mbox{I\hspace{-.14em}R}^3$ ), how would you compute the coefficients needed to write $\vec{y}$ as a linear combination of the basis vectors?
26.: How can you tell at a glance if a given matrix is orthogonally diagonalizable?
27.: What are the special properties of symmetric matrices?
28.: Given a quadratic form, how do you find a symmetric matrix A to write the quadratic form as x^T A x? If you were given A, what is the associated quadratic form?
29.: How do you find a change of variables to eliminate the cross terms in a quadratic form?

About this document ...

Tamara R. Olson
trolson at mtu.edu