Matrices
This operation flips a matrix over its main diagonal.
What is the transpose?
This method solves linear systems by eliminating variables using row operations.
What is Gaussian elimination?
A scalar λλ such that Av=λvAv=λv for some nonzero vv.
What is an eigenvalue?
This type of matrix has all zeros below the main diagonal.
What is an upper triangular matrix?
A linear transformation from a vector space to itself is called this.
What is an endomorphism (or linear operator)?
A matrix with this property satisfies AT=AAT=A.
What is symmetric?
A system with no solutions is called this.
What is an inconsistent system?
A matrix is diagonalizable if it has enough of these.
What are linearly independent eigenvectors?
A matrix whose determinant is zero is called this.
What is singular (or non-invertible)?
The matrix representation of this transformation scales every vector by a constant.
What is a scaling matrix (or homothety)?
This matrix, when multiplied by the original, gives the identity matrix.
What is the inverse matrix?
The number of pivots in a matrix’s row echelon form determines this.
What is the rank?
This decomposition writes A=PDP−1A=PDP−1, where DD is diagonal.
What is eigendecomposition?
This matrix has ones on the main diagonal and zeros elsewhere.
What is the identity matrix?
This property means a transformation preserves vector addition and scalar multiplication.
What is linearity?
The determinant of this special matrix is always 1, and its inverse is its transpose.
What is an orthogonal matrix?
A transformation TT is linear if it satisfies these two properties.
What are additivity (T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)) and homogeneity (T(cu)=cT(u)T(cu)=cT(u))?
For a symmetric matrix, these special eigenvectors are always orthogonal.
What are eigenvectors of a symmetric matrix?
A matrix is called this if AT=−AAT=−A.
What is skew-symmetric?
A transformation that preserves lengths and angles is called this.
What is an orthogonal transformation?
his decomposition writes a matrix as A=LUA=LU, where LL is lower triangular and UU is upper triangular.
What is LU decomposition?
This theorem states that for any matrix AA, rank(A)+nullity(A)=number of columnsrank(A)+nullity(A)=number of columns.
What is the Rank-Nullity Theorem?
This factorization, used in data science, breaks any matrix into UΣVTUΣVT.
What is Singular Value Decomposition (SVD)?
For a Markov matrix, all entries are non-negative, and columns sum to this number.
In Markov matrix, What is 1?
The determinant of a transformation matrix tells us this about its geometric effect.
What is the scaling factor of area/volume?