Final Exam (need to memorize)

det(u, v) + det(u, w)
det(u, w) + det(v, w)

k det(u, v) = det(ku, v)

The determinant of the permutation (det(P)) is…

• positive if even row exchanges

• negative if odd row exchanges

- det(v, u)

t * | a b | = t * det(matrix)
| c d |

There is linearity in the rows

zero

The determinant of the matrix doesn’t change, since the cofactor of the mutation evaluates to zero

zero

The determinant is the product of the diagonal entries

• product of pivots assuming no row exchanges

zero

• A matrix is invertible if it’s determinant does NOT equal zero

det(A) * det(B)

A(inverse) * A = Identity… therefore

• det(A(inv)) * det(A) = 1 (det of identity)

• det(A(inv)) = 1 / det(A)

det(A²) = det(AA) = det(A) * det(A)

det(2A) = 2^{n _*} det(A)

det(A)

Given a system Ax = b, Cramer’s rule gives a formula to find each component of x, such that

Cramer’s rule is only applicable IF A is an invertible matrix (det does not equal 0)

Matrix B_i is a matrix in which the ith column of A is replaced with the vector b and every other column is maintained

• x_i = det(B_i) / det(A)

Each entry of the C (cofactor) matrix is given by…

• c_ij = (-1)^{i + j} det(M_ij)

• M is the cofactor matrix

• solve the characteristic equation det(A - λI) = 0

• Each root is an eigenvalue of the matrix A

• find linearly independent vectors (a basis) of λ-eigenspace N(A - λI) for each eigenvalue λ

• The geometric multiplicity of an eigenvalue λ is dim N(A - λI), which is the number of linearly independent eigenvector associated with λ

det (A) = product of all eigenvalues

zero is an eigenvalue of the matrix

the number of non-zero eigenvalues is the same as the rank of the matrix A

(eigenvalues of A)^k

c(eigenvalue of A) + d for each eigenvalue of A

If and only if

• A has n linearly independent eigenvectors (x1, x2, . . . , xn)

If there exists and invertible matrix C such that…

• A = CBC^-1

These two matrices have the same eigenvalues (hence the same trace and determinants)

Every real symmetric matrix S can be factors into the form

• QΛQ^T, where Q is a real orthogonal matrix and Λ is a real diagonal matrix Λ

• Find the eigenvalues of the matrix S

• Find the eigenvectors to each eigenvalue value

  • If any two eigenvectors stem from the same eigenvalue… use gram Schmidt to find orthogonal vector of the two

• Normalize the eigenvectors (divide eigenvectors by magnitude of eigenvector)

• Each normalized eigenvector is in matrix Q and the diagonal matrix is made up of eigenvalues of S

1. All eigenvalues of S are positive

2. x^TSx > 0 for any non-zero vector x in Rn

3. S = A^TA for a full column rank matrix A (Cholesky Factorization)

4. All D_k > 0, where D_k is the determinant of the left-upper k by k submatrix of A

5. S has n pivots and all pivots are positive

1. All eigenvalues of S are non-negative

2. x^TSx >= 0 for any vector x in Rn

3. S = A^TA for a matrix A (Cholesky Factorization)

4. All D_k >= 0, where D_k is the determinant of the left-upper k by k submatrix of A

5. All pivots of S are positive