對於秩-1方陣 ,證明
。
Let be an
matrix and
. Prove that
.
證明一個可逆矩陣存在 QR 分解。
Prove that an invertible matrix can be represented in the form
, where
is an orthogonal matrix and
is an upper triangular matrix.
證明 Gram-Schmidt 正交化定理。
Let be a basis of an inner product space. Prove that there exists an orthogonal basis
such that
for all
.
證明 是不可逆矩陣的一個充分條件。
Let and
be
matrices, where
is an odd number. Prove that if
then at least one of the matrices
and
is singular.
這是零空間的包容關係與矩陣乘法的問題。
Let and
be complex matrices of size
and
, respectively. If
, prove that
for some
matrix
.
反對稱矩陣的伴隨矩陣 (adjugate) 是對稱或反對稱矩陣。
Let be an
skew-symmetric matrix. Prove that
is a symmetric matrix for odd
and a skew-symmetric matrix for even
.
這是關於基底的一個充分條件問題。
Let be vectors in
(
) such that
for
. Prove that any
of these vectors form a basis of
.
證明嚴格對角佔優 (strictly diagonally dominant) 矩陣是可逆矩陣。
Let be an
matrix. Prove that if
for
, then
is invertible.
證明矩陣積的值域與零空間的維數恆等式。
Let be an
matrix and
be an
matrix. Prove that
.
Note that and
denote the column space and nullspace of
, respectively.