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Matrix Space

Vector Interpretation​

Our Interpretation of vectors is something like this,

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Say we have a vector v⃗∈R3\vec{v}\in\mathbb{R}^3
then each element of vector v⃗\vec{v} represent an unique axis (say x,yx,y and zz).

  • these axis can be anything your salary, your SAT score, your running speed (anything which can be quantify in a number).*
    Assume a line passing thrown each element in that vector v⃗\vec{v}.
v⃗=[∙∙∙]\vec{v}=\begin{bmatrix} \color{red}{\bullet}\\ \color{blue}{\bullet}\\ \color{green}{\bullet}\\ \end{bmatrix}

interpret these ( red∙\color{red}{\text{red}\bullet}, blue∙\color{blue}{\text{blue}\bullet} and green∙\color{green}{\text{green}\bullet}) dots as real line ∈R1\in\mathbb{R}^1

Matrix Interpretation​

We can use our interpretation of vectors for matrices,

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Say we have a m×nm\times n matrix say A∈Rm×nA\in\mathbb{R}^{m\times n}
then each element of our matrix AA represent an unique axis (say u,v,w,⋯u,v,w,\cdots).

  • These axis can be anything your salary, your SAT score, your average running speed, average temperature in past 20 days, (anything which can be quantify in a number).*
    Assume a line passing thrown each element in this matrix AA.
A=[∙∙∙∙]A=\begin{bmatrix} \color{red}{\bullet} & \color{brown}{\bullet} \\ \color{green}{\bullet} & \color{blue}{\bullet} \\ \end{bmatrix}

interpret these (red∙\color{red}{\text{red}\bullet}, blue∙\color{blue}{\text{blue}\bullet} , green∙\color{green}{\text{green}\bullet} and brown∙\color{brown}{\text{brown}\bullet})

dots as real line ∈R1\in\mathbb{R}^1

Vector interpretation of a Matrix​

We can interpret matrices as vectors.

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Say we have a 2×22\times 2 matrix say A∈R2×2A\in\mathbb{R}^{2\times 2}

A=[∙∙∙∙]A=\begin{bmatrix} \color{red}{\bullet} & \color{brown}{\bullet} \\ \color{green}{\bullet} & \color{blue}{\bullet} \\ \end{bmatrix}

We can interpret it as vector say v⃗∈R4\vec{v}\in\mathbb{R}^4 as,

v⃗=[∙∙∙∙]\vec{v}=\begin{bmatrix} \color{red}{\bullet}\\ \color{brown}{\bullet}\\ \color{green}{\bullet}\\ \color{blue}{\bullet}\\ \end{bmatrix}

So we can see that our matrices are the vectors.

Matrix space

Think of all 3×33\times 3 matrices, there are not any limitation to elements, they can be anything, there are ∞\infty of them.

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These 3×33\times 3 matrices can we written as vectors v⃗∈R9\vec{v}\in\mathbb{R}^9

  • We can add those matrices, and if we add them we got another 3×33\times 3 matrix.
  • We can multiply matrices by a scaler, and if we multiply a scaler to a 3×33\times 3 matrix we get another 3×33\times 3 matrix.
  • We can take there linear combinations and the resultant matrix will still be a 3×33\times 3 matrix
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So the space of all 3×33\times 3 matrices is a vector space

Let's denote this space of all 3×33\times 3 matrices as M\mathcal{M}.
Now let's see sub spaces of M\mathcal{M}.

Lower Triangular​

All 3×33\times 3 Lower Triangular matrices ⊂\subset All 3×33\times 3 matrices.
Is the space of all 3×33\times 3 Lower Triangular matrices is a subspace of all 3×33\times 3 matrices?

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We know that All 3×33\times 3 Lower Triangular matrices ⊂\subset All 3×33\times 3 matrices, then,
Space of all Lower Triangular matrices is a subspace of all 3×33\times 3 matrices if,

  • All Lower Triangular 3×33\times 3 matrices form a vector space?

Do all Lower Triangular 3×33\times 3 matrices form a vector space?.

  • We can add two 3×33\times 3 Lower Triangular matrices, we got another 3×33\times 3 Lower Triangular matrix.
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[R00RR0RRR]+[R00RR0RRR]=[R00RR0RRR]\begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} + \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix}
  • We can multiply a 3×33\times 3 Lower Triangular matrices by a scaler, we get another 3×33\times 3 Lower Triangular matrix.
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C⋅[R00RR0RRR]=[R00RR0RRR]C\cdot \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix}
  • We can take there linear combinations and the resultant matrix will still be a 3×33\times 3 matrix
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α⋅[R00RR0RRR]+β⋅[R00RR0RRR]=[R00RR0RRR]\alpha\cdot \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} + \beta\cdot \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix}

So collectively we can say that all 3×33\times 3 Lower Triangular matrices forms a vector space.
So

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The space all 3×33\times 3 Lower Triangular matrices is a subspace of all 3×33\times 3 matrices .

Symmetric Matrix​

All 3×33\times 3 Symmetric matrices ⊂\subset All 3×33\times 3 matrices.
Is the space of all 3×33\times 3 Symmetric matrices is a subspace of all 3×33\times 3 matrices?

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We know that All 3×33\times 3 Symmetric matrices ⊂\subset All 3×33\times 3 matrices, then,
Space of all Symmetric matrices is a subspace of all 3×33\times 3 matrices if,

  • All Symmetric 3×33\times 3 matrices form a vector space?

Do all Symmetric 3×33\times 3 matrices form a vector space?.

  • We can add two 3×33\times 3 Symmetric matrices, we got another 3×33\times 3 Symmetric matrix.
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[RRRRRRRRR]+[RRRRRRRRR]=[2RR+RR+RR+R2RR+RR+RR+R2R]\begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} + \begin{bmatrix} \mathbb{R} & \color{brown}{\mathbb{R}} & \color{magenta}{\mathbb{R}} \\ \color{brown}{\mathbb{R}} & \mathbb{R} & \color{orange}{\mathbb{R}} \\ \color{magenta}{\mathbb{R}} & \color{orange}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} 2\mathbb{R} & \color{red}{\mathbb{R}}+\color{brown}{\mathbb{R}} & \color{blue}{\mathbb{R}}+\color{magenta}{\mathbb{R}} \\ \color{red}{\mathbb{R}}+\color{brown}{\mathbb{R}} & 2\mathbb{R} & \color{green}{\mathbb{R}}+\color{orange}{\mathbb{R}} \\ \color{blue}{\mathbb{R}}+\color{magenta}{\mathbb{R}} & \color{green}{\mathbb{R}}+\color{orange}{\mathbb{R}} & 2\mathbb{R} \\ \end{bmatrix}
  • We can multiply a 3×33\times 3 Symmetric matrices by a scaler, we get another 3×33\times 3 Symmetric matrix.
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C⋅[RRRRRRRRR]=[C⋅RC⋅RC⋅RC⋅RC⋅RC⋅RC⋅RC⋅RC⋅R]C\cdot \begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} C\cdot\mathbb{R} & \color{red}{C\cdot\mathbb{R}} & \color{blue}{C\cdot\mathbb{R}} \\ \color{red}{C\cdot\mathbb{R}} & C\cdot\mathbb{R} & \color{green}{C\cdot\mathbb{R}} \\ \color{blue}{C\cdot\mathbb{R}} & \color{green}{C\cdot\mathbb{R}} & C\cdot\mathbb{R} \\ \end{bmatrix}
  • We can take there linear combinations and the resultant matrix will still be a 3×33\times 3 matrix
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α⋅[RRRRRRRRR]+β⋅[RRRRRRRRR]=[α⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅Rα⋅R+β⋅R]\alpha\cdot \begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} + \beta\cdot \begin{bmatrix} \mathbb{R} & \color{brown}{\mathbb{R}} & \color{magenta}{\mathbb{R}} \\ \color{brown}{\mathbb{R}} & \mathbb{R} & \color{orange}{\mathbb{R}} \\ \color{magenta}{\mathbb{R}} & \color{orange}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \alpha\cdot\mathbb{R} + \beta\cdot\mathbb{R} & \color{red}{\alpha\cdot\mathbb{R}}+\color{brown}{\beta\cdot\mathbb{R}} & \color{blue}{\alpha\cdot\mathbb{R}}+\color{magenta}{\beta\cdot\mathbb{R}} \\ \color{red}{\alpha\cdot\mathbb{R}}+\color{brown}{\beta\cdot\mathbb{R}} & \alpha\cdot\mathbb{R} + \beta\cdot\mathbb{R} & \color{green}{\alpha\cdot\mathbb{R}}+\color{orange}{\beta\cdot\mathbb{R}} \\ \color{blue}{\alpha\cdot\mathbb{R}}+\color{magenta}{\beta\cdot\mathbb{R}} & \color{green}{\alpha\cdot\mathbb{R}}+\color{orange}{\beta\cdot\mathbb{R}} & \alpha\cdot\mathbb{R} + \beta\cdot\mathbb{R} \\ \end{bmatrix}

So collectively we can say that all 3×33\times 3 Symmetric matrices forms a vector space.
So

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The space all 3×33\times 3 Symmetric matrices is a subspace of all 3×33\times 3 matrices .

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Similarly The space all 3×33\times 3 Diagonal matrices is a subspace of all 3×33\times 3 matrices .