1P4 Mathematics Flashcards
What determines if a matrix is orthogonal?
if the inverse = transpose
What is a “proper orthogonal” matrix?
A matrix in which the determine is equal to one.
AKA Rotation matrix.
How do you determine the contents of a rotation matrix?
Consider where the basis vectors are rotated to. And assign these new vectors as the columns. Should form and orthoormal set.
What rotation is considered positive?
The right hand grip rule. Looking down the positive axis, anticlockwize.
How do you distinguish a rotation matrix and a reflection matrix?
Reflection matrix, the columns form a left handed set of unit vectors, and the detQ = -1 for rotation+reflection.
How do you determine the new coefficients in a rotated set of basis vectors?
Find the matrix that rotates a vector by that angle and transpose it to get the transformation to a new basis set.
How do you convert a matrix transformation to a new set of basis vectors?
A’ = RAR^T
Where R is the transformation to the new basis set.
How do you show that (AB) transpose = Bt At?
Express AB transpose in terms of the the sums of individual terms of and b. Show that these terms are the same as the swapped round version of Bt and At.
How do you find the inverse of a 3x3 matrix?
Find the cofactor matrix, alternating signs and matrix of minors.
Find the adjugate matrix by transposing the cofactor matrix.
inverse = 1/det x adjugate
What is true for matrix algebra?
Associative and distributive but not commutative.
How do you find the eigenvalues of a matrix?
det(A- lamba I) = 0
How do you find the eigenvectors of a matrix?
(A - λI) x = 0
What is true if S is a real, symmetrix matrix?
The eigenvalues and vectors are real. The eigen vectors are orthogonal.
What is an anti-symmetric matrix?
A^T = -A
Define a defective matrix.
An nxn matrix where it is impossible to find n linearly independent eigenvectors.
