18.085 :: Linear algebra cheat sheet :: Spring 2014
1. Rectangular m-by-nmatrices
•Rule for transposes:
(AB)T=BTAT
•The matrix ATAis always square, symmetric, and positive semi-definite. If in addition Ahas
linearly independent columns, then ATAis positive definite.
•Take Ca diagonal matrix with positive elements. Then ATCA is always square, symmetric, and
positive semi-definite. If in addition Ahas linearly independent columns, then ATC A is positive
definite.
2. Square n-by-nmatrices
•Eigenvalues and eigenvectors are defined only for square matrices:
Av =λv
The eigenvalues may be complex. There may be fewer than neigenvectors (in that case we say
the matrix is defective).
•In matrix form, we can always write
AS =SΛ,
where Shas the eigenvectors in the columns (Scould be a rectangular matrix), and Λis diagonal
with the eigenvalues on the diagonal.
•If Ahas neigenvectors (a full set), then Sis square and invertible, and
A=SΛS−1.
•If eigenvalues are counted with their multiplicity, then
tr(A) = A11 +. . . +Ann =λ1+. . . +λn.
•If eigenvalues are counted with their multiplicity, then
det(A) = λ1×. . . ×λn.
•Addition of a multiple of the identity shifts the eigenvalues:
λj(A+cI) = λj(A) + c.
No such rule exists in general when adding A+B.
•Gaussian elimination gives
A=LU.
Lis lower triangular with ones on the diagonal, and with the elimination multipliers below the
diagonal. Uis upper triangular with the pivots on the diagonal.
•Ais invertible if either of the following criteria is satisfied: there are nnonzero pivots; the deter-
minant is not zero; the eigenvalues are all nonzero; the columns are linearly independent; the
only way to have Au = 0 is if u= 0.
•Conversely, if either of these criteria is violated, then the matrix is singular (not invertible).
