Midterm 3 Review

BFITZ - Spring 2024

What is a unitary matrix?

Q* = Q^-1

What is a Hermition matrix

H* = H

The eigenvalues of A* are the…

conjugates of the eigenvalues of A

By the adjoint formula, <Av, w> =

<v, A*w>

det(A*)=

det(A)not

rank(A*) =

rank(A)

trace(A*) =

trace(A)not

(A^-1)*

c1notA1* + c2notA2*

What does conjugate transposition being an order-reversing involution imply?

(AB)* = B*A* and (A*)*

What is the conjugate transpose of A equal to?

A* = A^Tnot

What is right conjugate-linearity?

<v, cw> = c<v, w>

What is left conjugate-linearity?

<cv, w> = cnot<v, w>

What is the conjugate-symmetric property for the inner product in the complex plane?

<v, w> = <w, v>not

<v, w> = …?
What is the inner product of vectors in complex plane

Take the conjugate for each of the values of the FIRST vector, then use element-wise comparison

When you take the conjugate of a vector in the complex plane you have to…

take the conjugate of each of the values

What is the reciprocal of z=a+bi where z is not 0?

1/z = znot/(|z|²)

What is the absolute value of z = a+bi?

sqrt(a² + b²)

What is the conjugation of z = a+bi?

a-bi

What is the complex product for z1 and z2?

(a1a2 - b1b2) + (a1b2 + a2b1)i

Label the categories

Negative orientation, positive orientation, and coplanar

How do you determine the orientation and area of a parallelogram formed by vectors v and w?

The orientation is the SIGN of the det[v w], and the area is the ABSOLUTE VALUE of det[v and w]

Label the categories

Positive orientation, negative orientation, and colinear

What is Cramer’s Rule?

If A is invertible, then Ax = b is solved by x = (products of n det of Ai where each column i is replaced by b)^T/det(A)

Fill in the blank

adj(A)/det(A)

What does a matrix’s determinant say about its singularity?

If det(A) is 0, then A is singular

What is it = to?

det(A) x In

How are the cofactor and adjugate matrices of a matrix A related?

The adjugate matrix is the transpose of the cofactor matrix

In PA = LU, A = P^TLU. What do det(P^T), det(L), and det(U) equal?

det(P^T) = det(P) = (-1)^(# of swaps)
det(L) = diagonal product

det(U) = diagonal product

What is it equal to?

det(A)det(B)

What is it equal to?

det(A)

What is the determinant of a triangular matrix?

The product of its diagonal

DONT FORGET THIS!!

ith Row: aikCik
jth Col: akjCkj

DONT FORGET THIS!!

(-1)^even = +1

(-1)^odd = -1

What is a cofactor of A?

Cij = (-1)^(i+j) det(Aij)

What is a minor (i, j) of A?

Mij = det(Aij)

DONT FORGET THIS!!

determinant notation w/ sigma

Aij is obtained by____

deleting the ith row and jth column of A

What is R equal to in A = QR

R = Q^T A

How do you produce the columns of Q in A = QR?

Apply the Gram-Schmidt algorithm to the pivot columns of A

What is the gram-Schmidt general form for wd and qd?

wd = vd - projw1(vd)-…-projwd-1(vd)
qd = qd/||qd||

compv1(v) = ?

<v1, v>/||v1||

What does the projection onto a vector 1-D vector space reduce to?

v1<v1, v>/<v1^T, v1>

The least squares problem Axhat = Pb reduces to what in A = QR?

Rxhat = Q^tb

In A = QR, the projection onto Col(A) is equal to____

QQ^T

The columns of Q in A = QR form an orthonormal basis of ______

Col(A)

What do A and R have in common?

Common Gramian and rank

An m x r matrix Q has orthonormal columns if and only if_____

Q^TQ = Ir

What are the condition for a list of vectors to be orthonormal?

The vectors in the list are mutually orthogonal and each vector is a unit vector

What are the dimensions of Q and R? What is special about Q and R?

Q is m x r with orthogonal columns,

R is r x n upper-triangular

For a function f(rnot) = ?

f(r)not = 0

Quadratic form is q(x) = ?

<x, Sx>

What is the outcome of completing the square?

<y, Dy> → where y = U^Tx
lambda1y1² + lambda2y2² + lambda3y3²…

What is the definiteness of a Gramian matrix?

Positive semidefinite

How do you obtain the columns of U for H = UDU*

Apply Gram-Schmidt to the eigenspaces of H

What qualities to U and D have for a spectral factorization UDU*?

U is unitary and D is real-diagonal

A matrix H is Hermitian only if it has a….

Spectral factorization where H = UDU*

By the spectral theorem, every Hermitian matrix is…

diagonalizable

By the spectral theorem, the eigenspaces of a Hermitian matrix H are…

orthogonal to each other

Real-symmetric matrices are also known as…

Hermitian matrices

By the spectral theorem, the eigenvalues of every Hermitian matrix H are…

real

What is the diagonal rule for matrix exponentials?

e^diag(d1,…,dn) = diag(e^d1,…, e^dn)

e^(a+bi) = …?

e^a
{cos(b)+isin(b)}

If A is diagonalizable, then exp(A) = …?

Xexp(D)X^-1

u(t) = ?

u(t) = exp(At)u0

If A is diagonalizable, then the columns of X are the…

linearly independent eigenvectors of A

What are the two tests for diagonalizability?

The sum of each of the geometric multiplicities equals the size of the matrix OR the gm=am for each eigenvalue

(-1)^n (det(A)) = ?

the product of each of the eigenvalues raised to their algebraic multiplicities

trace(A) =

the sum of the products between a matrix’s eigenvalues and algebraic multiplicity

If A is diagonalizable, then the diagonal entries of D are…

the eigenvalues of A

If A is diagonalizable that means…

A is similar to a diagonal matrix

What is the power rule in A~B?

A^k = X(B^k)(X^-1)

What is the eigenspace rule of A~B?

EA(lambda) = XEB(lambda)

If A is similar to B, then A =

XBX^-1

Fill in the blanks

-trace(A) (-1)^n(det(A))

What are the geometric multiplicity bounds for the roots of the characteristic polynomial?

Between 1 and the am(lambda) inclusive

What do the algebraic multiplicities sum to?

the size of a matrix

The roots of xA(t) are the…

eigenvalues of A

xA^T(t) = ?

xA(t)

What is the characteristic polynomial xA(t)?

xA(t) = det(tIn - A)

What are the vieta formulas?

cn-1 = (m1r1 +….+ mkrk)
c0 = (-1)^n r1^m1… rk^mk

What does the sum of multiplicities equal to?

The degree of the polynomial

What is the form of the fundamental theorem of algebra?

f(t) = (t-r1)^m1…(t-rk)^mk