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Superposition Principle If is a solution to , and is a solution to , then is a solution to . Proof: Properties of Matrix Transpose 1) 2) 3) 4) Advertisements
Characteristic Polynomial, Eigenspace The solution space of is called the eigenspace of associated with the eigenvalue . The eigenspace is denoted by . Sum/Product of Eigenvalues – The sum of all eigenvalues of (including repeated eigenvalues) is the same as … Continue reading
Rotation Matrix The rotation matrix rotates points in the plane counterclockwise through an angle about the origin. For example rotating the vector 45 degrees counterclockwise gives us: Finding Least Squares Solution Given (inconsistent system), solve instead to get a least … Continue reading
Dot Product – – Span Subspaces is a subspace of if 1) for some vectors . 2) satisfies the closure properties: (i) for all , we must have . (ii) for all and , we must have . 3) is … Continue reading
Row echelon form (REF) For each nonzero row, the leading entry is to the right of the leading entry of the row above. E.g. Note that the leading entry 9 of the second row is to the right of the … Continue reading
This is an interesting question on vector subspaces (a topic from linear algebra): Question: If V and W are 2dimensional subspaces of , what are the possible dimensions of the subspace ? (A) 1 only (B) 2 only (C) 0 … Continue reading
Thanks for reading our blog Mathtuition88.com. We are also on: Facebook: https://www.facebook.com/mathtuition88 Twitter: https://twitter.com/mathtuition88 Google+: Mathtuition88 If you are a subscriber, thanks for your support! We will continue our mission of blogging interesting articles on Mathematics! Featured book: Coding the Matrix: Linear … Continue reading
Source: http://www.math.columbia.edu/~bayer/LinearAlgebra/pdf/inverse.pdf Backup copy (In case of broken link): 3×3 Matrix Inverse This is an excellent method for finding the inverse of a 3×3 matrix, probably the fastest and easiest method for 3×3. (Row reduction is better for 4×4 matrices and above.) … Continue reading
1) https://www.math.okstate.edu/~binegar/3013S99/3013l16.pdf (How to diagonalize matrices) 2) http://www.sosmath.com/matrix/expo/expo.html (How to find exponential of matrices) 3) http://en.wikipedia.org/wiki/Matrix_differential_equation (Matrix Differential Equations) 4) http://www.ucl.ac.uk/CNDA/courses/coursework/JNF.pdf (Jordan Normal Form) 5) http://www.math.niu.edu/~fletcher/jnf.pdf (Jordan Normal Form for 2×2 Matrices) 6) http://www.math.colostate.edu/~gerhard/M345/CHP/ch9_6.pdf (Matrix Exponential with repeated Eigenvalues) 7) http://www.math.vt.edu/people/afkhamis/class_home/notes/F08W12.pdf http://wwwusers.math.umn.edu/~leif0020/Spring2011/matrixexp.pdf (Matrix Differential equation with … Continue reading