4. Matrices Flashcards
For an element Amn, what does the m and n determine
M refers to the row of the element (height- think mountain)
N refers to column of the element (length- think Nile)
What is a vector?
A matrix with only one row or column
What is the trace?
It is the sum of the elements along the diagonal of a square matrix
Diagonal matrices
Square matrices with non zero entries only in the main diagonal and zero entries everywhere else
Identity matrix
A special type of diagonal matrix (In) with the number 1 in the main diagonal and zero everywhere else
Triangular matrix
Non zero entries in the positions above (below) the main diagonal and zero entries below (above) the main diagonal
Scalar
A matrix with a single entry (ie a real number)
When can you add and subtract matrices?
When they are of the same order
When can you multiply matrix Amxn and matrix Bpxq? And what dimensions will the new matrix have?
When n=p
The new matrix will have order m x q
What does B x In equal when B is a matrix?
B
Any matrix times by the identity matrix equals itself
What is a matrix transposition?
The rows and columns of the matrix are interchanged. It is essentially reflected in the line y=-x
Whet is the inverse matrix?
It is the matrix which satisfies the condition A x A^-1= A^-1A=In
What is 1/|A| x adj(A) equal to?
The inverse matrix A^-1
The determinant
|A| is a number which is a function of the entries of a square matrix
When can an inverse matrix be determined?
Only for a square matrix. Only if the determinant of the matrix doesn’t equal zero
What is the determinant of a matrix of order 1?
a11 (the only element available)
How is the determinant calculated?
By multiplying it’s elements by the corresponding cofactors and adding the products
Adjoint matrix
The transpose of the matrix obtained by replacing each element aij of matrix A with its corresponding cofactor
Does (AB)’ = B’A’
Yes
When is a matrix called singular?
When it doesn’t have an inverse
When are the inverses of matrices unique?
Always
What is (AB)^-1 equal to? Assuming AB is invertible
B^-1 x A^-1
What is (cA)^-1 equal to assuming c is a number not equal to 0
c^-1 x A^-1
What is the determinant used for?
To find the inverse matrix or determine if it exists
What is the sarrus rule?
A way of finding the determinant in a 3x3 matrix |A|= a11a22a33 + a12a23a31 + a13a21a32- a31a22a13- a32a23a11- a33a21a12
What is the minor of an element
|Aij| with deleted row i and column j is called the minor of element aij
What is the cofactor of an element?
The cofactor is the minor with the appropriate sign Cij= (-1)^(i+j) |Aij|
What is |A’| equal to?
|A|
What is |AB| equal to?
|A| x |B|
What is the way of finding the inverse of a particular element in a matrix?
Aij^-1= 1/|A| x Cij
What can be said about the equation system Ax=b if |A| is not equal to zero?
There is an unique inverse A^-1. The solution x=A^-1 x b is the only solution
When dealing with eigenvalues what is k?
An unknown n element column vector
What is r
An unknown scalar
When do we get the trivial solution k=0
When A-rI is non singular
When do we get a non trivial solution to k?
When A-rI is singular so the determinant of A-rI=0
Eigenvalues
The values of r that are a solution to det(A-rI)=0
Eigenvector
A nonzero vector ki which is a particular solution of the equation for a particular eigenvalue ri
Whet does the trace of matrix A equal?
The sum of the eigenvalues of A
What does the product of the eigenvalues of A equal?
The determinant of A
When is a quadratic form q(x) positive definite?
If q(x)>0 for all x=\ 0
When is a quadratic form q(x) positive semi definite?
If q(x)>= for all x =\0
When is a quadratic form q(x) negative definite?
If q(x)<0 for all x =\0
When is a quadratic form q(x) negative semi definite?
If q(x)<=0 for all x=\0
How can whether a quadratic form is +ve or -ve definite be checked
By looking at its leading principal minors I.e. the determinants of the leading principal is matrices of A
When is a symmetric matrix positive definite?
If every leading principal minor is positive
When is a symmetric matrix negative definite?
If the leading principal minors alternate in sign starting with negative
When is a symmetric matrix positive semi definite?
Determinants of all principal submatrices >=0
When is a symmetric matrix negative semi definite
Determinants of all principal submatrices of odd order <=0 and determinants of all principal submatrices of even order >=0
Using eigenvalues when is a symmetric matrix +ve definite?
If all eigenvalues are +ve
Using eigenvalues when is a symmetric matrix +ve semidefinite?
If all eigenvalues are non -ve
Using eigenvalues when is a symmetric matrix -ve definite?
If all eigenvalues are -ve
Using eigenvalues when is a symmetric matrix -ve semidefinite?
If all eigenvalues are non positive
Using eigenvalues when is a symmetric matrix indefinite?
If it has a +ve and -ve eigenvalues
