MA511
spectral theorem
D=U^TAU
All fourier matrices are Hermitian
False
Rank(AB) >= (rank(A), rank(B))
False
Suppose A is a real symmetric negative definite matrix of size 4x4 that is x^TAx<0: All upper left minors of A are negative
False
The row space of a square matrix is orthogonal to the column space
False
if A is nxn matrix and A^2=0 then A, I+A, and I-A are all invertible
False
The differential equation du/dt = Au is stable if Re(lambdai) > 0 for all eigenvalues.
False Condition 1: a+b must be negative Condition 2: the determinant must be positive
If the characteristic polynomials of 2x2 matrices A and B are equal then A and B are similar
False:
If A is an mxn and B is nxr matrix then N(AB) >= N(A)
False: AB = mxr : N(AB) -> R^r A = mxn : N(A) ->R^n the subspaces are different dimensions so this is nonsense
If A is an nxn matrix and rank (A^TA)<n then rank(A)<n
False: If A is invertible then rank(A)=n
If V and W are subspaces of R^11 and dim(V)=5 and dim(W)=8 then dim(V intersect W)>=4
False: dim(V intersect U) = dimV + dimW - dim(V+W) 5+8-11
If A and B are unitary then A+B is unitary
False: Example: A=I B=-I. They are unitary but their sum is zero, not unitary
If U is a unitary matrix then all its eigenvalues are real
False: example [i]; this matrix is unitary and not real
If A and B are Hermitian, then A*B is Hermitian
False: A*B*=B*A* This is equal to A*B* only when A and B commute A+B is hermitian A*B+B*A is hermitian
Let A be an nxn matrix. If A is non-singular then it is diagnoalizable
False: Counter example is a Jordan Cell with non-zero eigenvalue
Suppose A is a real symmetric negative definite matrix of size 4x4 that is x^TAx<0: Determinant of A is negative
False: Determinant is the product of eigenvalues. Since they are all negative, and there are 4 of them, the determinant is positive
If A and B are nxn matrices of the same size, e^(A+B)=e^A*e^B
False: The left hand side does not change if we interchange A and B while the right hand side can change (order of matrices matters)
For a 5x5 matrix A, det(3A) = 3det(A)
False: the correct formula is det(3A) = 3^5det(A)
All fourier matrices Fn are unitarily diagonalizable
True
Given an mxn matrix A, the equation Ax=0 has ONLY a trivial solution if its columns are linearly independent
True
If A and B are both Hermitian matrices, then A+B is Hermitian
True
If A is Hermitian and U is unitary of the same size then U^-1AU is Hermitian
True
If A is a square matrix such that the linear transformation f(x)=Ax, preserves angles and lengths then A^TA is an orthogonal matrix
True
If A is singular then it has eigenvalue 0
True
If P is a projection matrix and P is invertible then P = I
True
If x^tA=b^T has exactly one solutions then rows of A are linearly independent
True
The set of all symmetric nxn matrices is a vector space
True
for a square matrix A the following four properties are equivalent: 1. A is left-invertible 2. the columns of A are linearly independent 3. A is right-invertible 4. the rows of A are linearly independent
True
if A is an nxn matrix and the system Ax=b has a unique solution, then A is a product of elementary matrices
True
If {q1, q2, q3} is an orthonormal set of vectors then q1q1^T+q2q2^T+q3q3^T is a projection matrix
True P=QQ^T
If P^2 = P and P=P^T then P is a projection matrix
True! this is definition of projection
A consistent linear system of 4 equations in 5 unknowns has infinitely many solutions
True:
If A and B are unitary matrices with the same eigenvalues then there is unitary U such that UBU^H=A
True:
If A is symmetric then RAR^T is symmetric
True: (RAR^T)^T = R^TA^TR and A=A^T bc it is symmetric
If A is symmetric positive definite then A=B^4 for some B which is symmetric positive definite
True: If A is positive definite there exists A=B^2 If B is positive definite there exists B^2=C^4 and so on
If U1 and U2 are unitary matrices of the same size, then U1U2^H is unitary
True: The product of two unitary matrices are unitary
If U is a unit vector then I - uu^T is a projection matrix
True: To be a projection: 1. must be symmetric 2. squared is equal to itself
If A is a square matrix which has a left inverse then A^TA is nonsingular
True: for a square matrix: nonsingular=invertible
For a 5x5 matrix A, det(-A) = -det(A)
True: (-1)^5 = -1
If A and B are unitary then AB is unitary
True: (AB)(AB)*=ABB*A*=I
Suppose A is a real symmetric negative definite matrix of size 4x4 that is x^TAx<0: no row exchanges are required when bringing A to upper triangular form by row operations
True: -A is positive definite For positive definite matrix, no row exchanges are required. Thus the same is for A
Suppose A is a real symmetric negative definite matrix of size 4x4 that is x^TAx<0: All eigenvalues of A are strictly negative
True: By the spectral theorem, A is congruent to the diagonal matrix with eigenvalues on the main diagonal.
for an nxn matrix A, if A^2019=0 then A is singular
True: If A^2019x=0, x =/0, then A^2018x belongs to N(A). If A^2018x=0 then A^2017x belongs to N(A) and so on. We eventually find a non-zero vector in N(A). This means A is singular.
If A is Hermitian and B is unitary then B^-1AB is defined as Hermitian
True: This is defined because unitary matrices are invertible, and B^-1=B* (B^-1AB)* = (B*AB)* = B*A*B = B^-1AB
Let A and B be the same size nxn matrices. If A and B are similar, then trA=trB
True: similar matrices have the same characteristic polynomial
For a 5x5 matrix A: the determinant of A does not change if the rows A are rearranged in the opposite order
True: this depends on the sign of the permutation. for n=5, there are two transposes thus it is even.
Suppose A is a real symmetric negative definite matrix of size 4x4 that is x^TAx<0: A is non-singular
True: x^tAx<0 for all x not equal to zero means that the signature consists of n minuses
If Q has orthonormal rows then QQ^T=I
True; orthonormal rows requires the matrix to be square. Thus this works
If P is an nxn projection matrix then rank(P)=rank(I-P)
false
If a is a non-zero vector in R^n, and P is the projection matrix for the orthogonal projection on to span(a), then the rank(P) = n-1
false
if P is an nxn projection matrix, then rank(P)=n
false
if U is an orthogonal matrix, then U=U^T
false
suppose that A and B are matrices of the same size then, rank(A+B)<= min(rank(A),rank(B))
false
suppose that a homogeneous system of m equations in n unknowns has no nontrivial solution. then m=n
false
the row space of a square matrix is orthogonal to the column space
false
the set of orthogonal matrices form a vector space
false
The set of all non-invertible nxn matrices is a vector space
false: zero is included, sum does not work
If A and B are the same size nxn matrices, det(A+B)=det(A)+det(B)
false: ex Identity matrix det(I+I)=0 det(I)+det(I)=2
What is hermitian
matrix A = the conjugate transpose of A
A singular matrix is a square matrix that has no inverse
true
For an nxn matrix A, if v is an eigenvector of A the v is also an eigenvector of e^A
true
For square nxn matrices A, B, and C; det(ABC)=det(A)det(B)det(C)
true
If A has infinitely many inverse solutions then it has no right inverse
true
If A is hermitian and U is unitary of the same size then U^-1AU is hermitian
true
If A is mxn and rank(A)=m then A has linearly independent rows
true
If P is an nxn projection matrix then I-P is also a projection
true
If U is an orthogonal matrix, then U is invertible
true
If U is an orthogonal matrix, then U^T is also orthogonal
true
If v is an eigenvector of A then it is an eigenvector of A+A^2
true
Suppose that A and B are nxn square matrices and A is invertible. Then, nullity(AB)=nullity(BA)
true
The column space is orthogonal to the left nullspace
true
The row space of a matrix is orthogonal to the null space
true
A homogeneous system of four equations in fie unknowns has a non-trivial solution
true dimN(A) + rank(A)= # of cols
If u1 and u2 are solutions to the nonhomogeneous system Ax=b, then u1 - u2 is also a solution to Ax=0
true: Au1=b Au2=b A(u1-u2)= Au1 - Au2 = b - b = 0
If A is Hermitian and real then A is symmetric
true: If A is real then the conjugate of A = A. A = A^H (definition of Hermitian) ITherefore, A = A^T So it is confirmed symmetric
If A is orthogonal and symmetric then A has eigenvalues of 1, and -1
true: eigenvalues of symmetric matrices are real and eigenvalues of orthogonal matrix have unit modulus